title: Bilbao

text: The average amount of time people spend commuting with public transit in Bilbao, for example to and from work, on a weekday is 35 minutes. Three percent of public transit passengers ride for more than 2 hours every day. The average amount of time people wait at a stop or station for public transit is 7 minutes, while 4% of riders wait for over 20 minutes on average every day. The average distance people usually ride in a single trip with public transit is , while 10% travel more than in a single direction.


title: Bilbao

text: The public healthcare system is served and managed by the Basque healthcare system ("Osakidetza"), which has a network of medical centres established across the city. The most relevant hospital of the city is the Basurto Hospital, located in the Basurto-Zorroza district. It was opened in 1908 after ten years of construction, and it was built due to the fast increase of population in the late 19th century. Its design was based on the Eppendorf Hospital of Hamburg, in (Germany), one of the most modern at the time. In 2008 the hospital was named the best hospital in customer support in Spain.


title: Bilbao

text: However, the two largest hospitals that provide healthcare services to the inhabitants of the city are located outside of the city limits: the University Hospital of Gurutzeta/Cruces in neighboring Barakaldo and the Galdakao-Usansolo Hospital in Galdakao. Other minor public hospitals located in the city include the Santa Marina Hospital and the Hospital of Urduliz, located in the metropolitan area.


title: Bilbao

text: The headquarters of "Iberdrola", a multinational electric utility company are located in Bilbao. The company was created in 1992 after the merging of "Iberduero" and "Hidroeléctrica Española" and it is in charge of production and distribution of electricity. There are two thermal power stations located in the city's surroundings, one in Santurtzi and the other in Zierbena. The electricity produced in both these stations is then transferred to the electrical substation of Güeñes by the company "Red Eléctrica de España", which is the company responsible of moving electricity from the production stations to the final consumers. There is a combined cycle power station in the municipality of Amorebieta called "Bizkaia Energia", whose energy is transferred to the substation of Gatika.


title: Bilbao

text: The supply of all oil-based fuels distributed in the whole metropolitan area (gasoline, diesel fuel and butane) are produced in the oil refinery owned by "Petronor" in the municipalities of Muskiz and Abanto Zierbena. "Petronor" is a large oil company founded in Bilbao in 1968 and it currently comprises two partners, Repsol (85.98%) and Kutxabank (14.02%).


title: Bilbao

text: Supply of drinking water is managed by the "Consorcio de Aguas Bilbao Bizkaia" (Bilbao Bizkaia Water Consortium), a public entity with the responsibility of managing the water infrastructure and distribution to many Biscayan municipalities, Bilbao and its metropolitan area included. The supply of drinking water for the city comes from the Uribarri-Ganboa and Urrunaga reservoirs, which are fed by the river Zadorra.


title: Bilbao

text: The Basque Country has a bilingual education system, with students able to choose between four linguistic models: A, B, D, and X, which differ in the prevalence of Basque or Spanish as the spoken and written language used in classes. In Bilbao, there is a prevalence of model D (where Basque is the vehicle language and Spanish is taught as a subject) in Primary School, while Compulsory Secondary Education students favour model B (where some subjects are in Basque and other in Spanish). Finally, 67% of Baccalaureate students choose model A (in which Spanish is the vehicle language and Basque is a subject). English is the most widespread foreign language taught, being the option for 97% of pre-university students.


title: Bilbao

text: Two universities are seated in Bilbao. The older is the University of Deusto, founded by the Society of Jesus in 1886. It took its name from the then independent municipality of Deusto, annexed to Bilbao in 1925. It was the only higher education institute in the borough until the establishment in 1968 of the University of Bilbao, later to become the University of the Basque Country in 1980. This public university, which has a presence in the three provinces of the autonomous community, has its main Biscayan campus in the municipality of Leioa, although the Technical and Business faculties are based in Bilbao. Since 2014, Mondragon University has also a presence in the city through the innovation and entrepreneurial centre Bilbao Innovation Factory. In 2015, the offer of higher education in the city was expanded with the foundation of Dantzerti, the Higher School of Dramatic Arts and Dance of the Basque Country.


title: Bilbao

text: As in the rest of Spain, football is the most popular competitive sport, followed by basketball. Some people are interested in basque sports, one of the oldest sports in Spain. One could enjoy a Basque pelota match in Bizkaia Frontoia, Bilbao. In addition, Bilbao offers many outdoor activities owing to its location by the sea amid the hilly countryside. Hiking and rock climbing in the nearby mountains are very popular. Watersports, especially surfing, are practised on the beaches of nearby Sopelana and Mundaka.


title: Bilbao

text: The main football club is Athletic Club, commonly known as Athletic Bilbao in English. It plays at the new San Mamés stadium, which opened in 2013 and seats 53,332 spectators.


title: Bilbao

text: The main basketball club is Bilbao Basket, which plays in the Spanish top division Liga ACB and their home venue is the Bilbao Arena. The main achievements of the clubs were being the runner-up of the Liga ACB during the 2010–2011 season, losing the final series 3–0 against FC Barcelona Bàsquet; as well as the runner-up of the EuroCup Basketball in the 2012–2013 edition, where they lost to the Russian team Lokomotiv Kuban.


title: Patroclus

text: In Greek mythology, as recorded in Homer's "Iliad", Patroclus (pronunciation variable but generally ; ) was a childhood friend, close wartime companion, and the presumed (by some later ancient sources) lover of Achilles.


title: Patroclus

text: There are at least three pronunciations of the name 'Patroclus' in English. Because the penultimate syllable is light in Latin prose ("pă′.trŏ.clŭs"), the antepenult was stressed in Latin and would normally be stressed in English as well, for (analogous to 'Sophocles'). However, this pronunciation is seldom encountered: for metrical convenience, Alexander Pope had made the 'o' long, and thus stressed, in his translation of Homer, following a convention of Greek and Latin verse, and that pronunciation – of Latin "pa.trō′.clus" – has stuck, for English .


title: Patroclus

text: Moreover, because in prose a penultimate Greco-Latin short "o" (omicron) would only be stressed in a closed syllable, the penult has sometimes been misanalysed as being closed (*pă.trŏc′.lŭs), which would change the English "o" to a short vowel: .


title: Patroclus

text: In the account of Dares the Phrygian, Patroclus was illustrated as ". . .handsome and powerfully built. His eyes were gray. He was modest, dependable, wise, a man richly endowed."


title: Patroclus

text: Patroclus was the son of Menoetius by either Philomela or Polymele, Sthenele, Periopis, or lastly Damocrateia. His only sibling was Myrto, mother of Eucleia by Heracles. Homer also references Menoetius as the individual who gave Patroclus to Peleus. Menoetius was the son of Actor, king of Opus in Locris by Aegina, daughter of Asopus.


title: Patroclus

text: During his childhood, Patroclus had killed another child in anger over a game. Menoetius gave Patroclus to Peleus, Achilles' father, who named Patroclus Achilles' "squire" as Patroclus and Achilles grew up together, and became close friends. Patroclus acted as a male role model for Achilles, as he was both kinder than Achilles and wise regarding counsel.


title: Patroclus

text: According to Ptolemaeus Chennus, Patroclus also became the lover of the sea-god Poseidon, who taught him the art of riding horses.


title: Patroclus

text: According to the "Iliad", when the tide of the Trojan War had turned against the Greeks and the Trojans were threatening their ships, Patroclus convinced Achilles to let him lead the Myrmidons into combat. Achilles consented, giving Patroclus the armor Achilles had received from his father, in order for Patroclus to impersonate Achilles. Achilles then told Patroclus to return after beating the Trojans back from their ships. Patroclus defied Achilles' order and pursued the Trojans back to the gates of Troy. Patroclus killed many Trojans and Trojan allies, including a son of Zeus, Sarpedon. While fighting, Patroclus' wits were removed by Apollo, after which Patroclus was hit by the spear of Euphorbos. Hector then killed Patroclus by stabbing him in the stomach with a spear.


title: Patroclus

text: Achilles retrieved his body, which had been stripped of armor by Hector and protected on the battlefield by Menelaus and Ajax. Achilles did not allow the burial of Patroclus' body until the ghost of Patroclus appeared and demanded his burial in order to pass into Hades. Patroclus was then cremated on a funeral pyre, which was covered in the hair of his sorrowful companions. As the cutting of hair was a sign of grief while also acting as a sign of the separation of the living and the dead, this points to how well-liked Patroclus had been. The ashes of Achilles were said to have been buried in a golden urn along with those of Patroclus by the Hellespont.


title: Patroclus

text: Although there is no explicit sexual relationship between Achilles and Patroclus in the Homeric tradition, a few later Greek authors wrote about what they saw as implied in the text regarding their relationship. Aeschylus and Phaedrus, for example, state there was a clear relationship between them, and they both refer to Achilles as the eromenos of the relationship. Morales and Mariscal state "there is a polemical tradition concerning the nature of the relationship between the two heroes". According to Ledbetter (1993), there is a train of thought that Patroclus could have been a representation of the compassionate side of Achilles, who was known for his rage, mentioned in the first line of Homer's "Iliad". Ledbetter connects the way that Achilles and his mother Thetis communicate to the link between Achilles and Patroclus. Ledbetter does so by comparing how Thetis comforts the weeping Achilles in Book 1 of the "Iliad" to how Achilles comforts Patroclus as he weeps in Book 16. Achilles uses a simile containing a young girl tearfully looking at her mother to complete the comparison. Ledbetter believes this puts Patroclus into a subordinate role to that of Achilles. However, as Patroclus is explicitly stated to be the elder of the two characters, this is not evidence of their ages or social relation to each other.


title: Patroclus

text: James Hooker describes the literary reasons for Patroclus' character within the "Iliad". He states that another character could have filled the role of confidant for Achilles, and that it was only through Patroclus that we have a worthy reason for Achilles' wrath. Hooker claims that without the death of Patroclus, an event that weighed heavily upon him, Achilles' following act of compliance to fight would have disrupted the balance of the "Iliad". Hooker describes the necessity of Patroclus sharing a deep affection with Achilles within the "Iliad". According to his theory, this affection allows for the even deeper tragedy that occurs. Hooker argues that the greater the love, the greater the loss. Hooker continues to negate Ledbetter's theory that Patroclus is in some way a surrogate for Achilles; rather, Hooker views Patroclus' character as a counterpart to that of Achilles. Hooker reminds us that it is Patroclus who pushes the Trojans back, which Hooker claims makes Patroclus a hero, as well as foreshadowing what Achilles is to do.


title: Patroclus

text: Achilles and Patroclus grew up together after Menoitios gave Patroclus to Achilles' father, Peleus. During this time, Peleus made Patroclus one of Achilles' "henchmen". While Homer's "Iliad" never explicitly stated that Achilles and Patroclus were lovers, this concept was propounded by some later authors. Aeschines asserts that there was no need to explicitly state the relationship as a romantic one, for such "is manifest to such of his hearers as are educated men." In later Greek writings, such as Plato's "Symposium", the relationship between Patroclus and Achilles is discussed as a model of romantic love. However, Xenophon, in his "Symposium", had Socrates argue that it was inaccurate to label their relationship as romantic. Nevertheless, their relationship is said to have inspired Alexander the Great in his own close relationship with his life-long companion Hephaestion.


title: Patroclus

text: Achilles was younger than Patroclus. This reinforces Dowden's explanation of the relationship between an eromenos, a youth in transition, and an "erastes", an older male who had recently made the same transition. Dowden also notes the common occurrence of such relationships as a form of initiation.


title: Patroclus

text: Patroclus is a character in William Shakespeare's play "Troilus and Cressida". In the play Achilles, who has become lazy, is besotted with Patroclus, and the other characters complain that Achilles and Patroclus are too busy having sex to fight in the war.


title: Ruby Bridges

text: Ruby Nell Bridges Hall (born September 8, 1954) is an American civil rights activist. She was the first African-American child to desegregate the all-white William Frantz Elementary School in Louisiana during the New Orleans school desegregation crisis on November 14, 1960. She is the subject of a 1964 painting, "The Problem We All Live With", by Norman Rockwell.


title: Ruby Bridges

text: Bridges was the eldest of five children born to Abon and Lucille Bridges. As a child, she spent much time taking care of her younger siblings, though she also enjoyed playing jump rope, softball and climbing trees. When she was four years old, the family relocated from Tylertown, Mississippi, where Bridges was born, to New Orleans, Louisiana. In 1960, when she was six years old, her parents responded to a request from the National Association for the Advancement of Colored People (NAACP) and volunteered her to participate in the integration of the New Orleans school system, even though her father was hesitant.


title: Ruby Bridges

text: Bridges was born during the middle of the Civil Rights Movement. "Brown v. Board of Education" was decided three months and twenty-two days before Bridges' birth. The court ruling declared that the establishment of separate public schools for white children, which black children were barred from attending, was unconstitutional; accordingly, black students were permitted attend such schools. Though the "Brown v. Board of Education" decision was finalized in 1954, southern states were extremely resistant to the decision that they must integrate within six years. Many white people did not want schools to be integrated and, though it was a federal ruling, state governments were not doing their part in enforcing the new laws. In 1957, federal troops were ordered to Little Rock, Arkansas, to escort the Little Rock Nine students in combating violence that occurred as a result of the decision. Under significant pressure from the federal government, the Orleans Parish School Board administered an entrance exam to students at Bridges' school with the intention of keeping black children out of white schools.


title: Ruby Bridges

text: Bridges attended a segregated kindergarten in 1959. In early 1960, Bridges was one of six black children in New Orleans to pass the test that determined whether they could go to the all-white William Frantz Elementary School. Two of the six decided to stay at their old school, Bridges went to Frantz by herself, and three children were transferred to McDonogh No. 19 and became known as the McDonogh Three. Bridges and her mother were escorted to school by four federal marshals during the first day that Bridges attended William Frantz Elementary. In the following days of that year, federal marshals continued to escort Bridges, though her mother stayed behind to take care of her younger siblings.


title: Ruby Bridges

text: Bridges' father was initially reluctant, but her mother felt strongly that the move was needed not only to give her own daughter a better education, but to "take this step forward ... for all African-American children". Her mother finally convinced her father to let her go to the school.


title: Ruby Bridges

text: Judge J. Skelly Wright's court order for the first day of integrated schools in New Orleans on Monday, November 14, 1960, was commemorated by Norman Rockwell in the painting, "The Problem We All Live With" (published in "Look" magazine on January 14, 1964). As Bridges describes it, "Driving up I could see the crowd, but living in New Orleans, I actually thought it was Mardi Gras. There was a large crowd of people outside of the school. They were throwing things and shouting, and that sort of goes on in New Orleans at Mardi Gras." Former United States Deputy Marshal Charles Burks later recalled, "She showed a lot of courage. She never cried. She didn't whimper. She just marched along like a little soldier, and we're all very very proud of her."


title: Ruby Bridges

text: As soon as Bridges entered the school, white parents pulled their own children out; all the teachers except for one refused to teach while a black child was enrolled. Only one person agreed to teach Bridges and that was Barbara Henry, from Boston, Massachusetts, and for over a year Henry taught her alone, "as if she were teaching a whole class."


title: Ruby Bridges

text: That first day, Bridges and her mother spent the entire day in the principal's office; the chaos of the school prevented their moving to the classroom until the second day. On the second day, however, a white student broke the boycott and entered the school when a 34-year-old Methodist minister, Lloyd Anderson Foreman, walked his five-year-old daughter Pam through the angry mob, saying, "I simply want the privilege of taking my child to school ..." A few days later, other white parents began bringing their children, and the protests began to subside. Yet, still, Bridges remained the only child in her class, as she would until the following year. Every morning, as Bridges walked to school, one woman would threaten to poison her, while another held up a black baby doll in a coffin; because of this, the U.S. Marshals dispatched by President Eisenhower, who were overseeing her safety, allowed Bridges to eat only the food that she brought from home.


title: Ruby Bridges

text: Child psychiatrist Robert Coles volunteered to provide counseling to Bridges during her first year at Frantz. He met with her weekly in the Bridges home, later writing a children's book, "The Story of Ruby Bridges", to acquaint other children with Bridges' story. Coles donated the royalties from the sale of that book to the Ruby Bridges Foundation, to provide money for school supplies or other educational needs for impoverished New Orleans school children.


title: Ruby Bridges

text: The Bridges family suffered for their decision to send her to William Frantz Elementary: her father lost his job as a gas station attendant; the grocery store the family shopped at would no longer let them shop there; her grandparents, who were sharecroppers in Mississippi, were turned off their land; and Abon and Lucille Bridges separated. Bridges has noted that many others in the community, both black and white, showed support in a variety of ways. Some white families continued to send their children to Frantz despite the protests, a neighbor provided her father with a new job, and local people babysat, watched the house as protectors, and walked behind the federal marshals' car on the trips to school. It was not until Bridges was an adult that she learned that the immaculate clothing she wore to school in those first weeks at Frantz was sent to her family by a relative of Coles. Bridges says her family could never have afforded the dresses, socks, and shoes that are documented in photographs of her escort by U.S. Marshals to and from the school.


title: Ruby Bridges

text: Bridges, now Ruby Bridges Hall, still lives in New Orleans with her husband, Malcolm Hall, and their four sons. After graduating from a desegregated high school, she worked as a travel agent for 15 years and later became a full-time parent. She is now chair of the Ruby Bridges Foundation, which she formed in 1999 to promote "the values of tolerance, respect, and appreciation of all differences". Describing the mission of the group, she says, "racism is a grown-up disease and we must stop using our children to spread it."


title: Ruby Bridges

text: Bridges is the subject of the Lori McKenna song "Ruby's Shoes". Her childhood struggle at William Frantz Elementary School was portrayed in the 1998 made-for-TV movie "Ruby Bridges". The young Bridges was portrayed by actress Chaz Monet, and the movie also featured Lela Rochon as Bridges' mother, Lucille "Lucy" Bridges; Michael Beach as Bridges' father, Abon Bridges; Penelope Ann Miller as Bridges' teacher, Mrs. Henry; and Kevin Pollak as Dr. Robert Coles.


title: Ruby Bridges

text: Like hundreds of thousands of others in the greater New Orleans area, Bridges lost her home (in Eastern New Orleans) to catastrophic flooding from the failure of the levee system during Hurricane Katrina in 2005. Hurricane Katrina also greatly damaged William Frantz Elementary School, and Bridges played a significant role in fighting for the school to remain open.


title: Ruby Bridges

text: In November 2007, the Children's Museum of Indianapolis unveiled a new permanent exhibit documenting her life, along with the lives of Anne Frank and Ryan White. The exhibit, called "The Power of Children: Making a Difference", cost $6 million to install and includes an authentic re-creation of Bridges' first grade classroom.


title: Ruby Bridges

text: In 2010, Bridges had a 50th-year reunion at William Frantz Elementary with Pam Foreman Testroet, who had been, at the age of five, the first white child to break the boycott that ensued from Bridges' attendance at that school.


title: Ruby Bridges

text: On July 15, 2011, Bridges met with President Barack Obama at the White House, and while viewing the Norman Rockwell painting of her on display he told her, "I think it's fair to say that if it hadn't been for you guys, I might not be here and we wouldn't be looking at this together". The Rockwell painting was displayed in the West Wing of the White House, just outside the Oval Office, from June through October 2011.


title: Ruby Bridges

text: In September 1995, Bridges and Robert Coles were awarded honorary degrees from Connecticut College and appeared together in public for the first time to accept the awards.


title: Ruby Bridges

text: On August 10, 2000, the 40 year anniversary of her walk into William Frantz Elementary School, Deputy Attorney General Eric Holder made Ruby Bridges an Honorary Deputy U.S. Marshal.


title: Ruby Bridges

text: In November 2006, Bridges was honored as a "Hero Against Racism" at the 12th annual Anti-Defamation League "Concert Against Hate" with the National Symphony Orchestra, held at the Kennedy Center in Washington, DC.


title: Ruby Bridges

text: On May 19, 2012, Bridges received an Honorary Degree from Tulane University at the annual graduation ceremony at the Superdome.


title: Ruby Bridges

text: Two elementary schools are named after Bridges: one in Alameda, California, and another in Woodinville, Washington. A statue of Bridges stands in the courtyard of William Frantz Elementary School.


title: Horchata

text: Horchata (; ), or (), is a name given to various beverages, which are generally plant-based, but sometimes contain animal milk. In Spain, it is made with soaked, ground, and sweetened tiger nuts. In Latin America and other parts of the Americas, the base is jicaro, melon or sesame seeds, or white rice, along with other spices. Different varieties can be served hot or cold, and may be used as a flavor in other beverages, such as frappé coffee.


title: Horchata

text: The name probably derives from a Latin word for barley, the term , which in turn comes from (barley), related to a Mediterranean tradition of grain-based beverages. The Italian and Maltese , the French and English "orgeat" have the same origin, although the beverages themselves have diverged, and are generally no longer made from barley.


title: Horchata

text: The drink originated in North Africa, and it is estimated that during the 11th century, it began to spread throughout Hispania (now Spain and Portugal). There are 13th-century records of a -like beverage made near Valencia, where it remains a common drink.


title: Horchata

text: From Spain, the concept of horchata was brought to the New World. Here, drinks called or simply came to be made with white rice and cinnamon or instead of tiger nuts. Sometimes these drinks had vanilla added, or were served adorned with fruit. Similarly flavored plant based beverages are sold in various parts of the world as varieties of or .


title: Horchata

text: The drink now known as (also sometimes called or, in West African countries such as Nigeria and Mali, ) is the original form of horchata. It is made from soaked, ground and sweetened tiger nuts.<ref name="spainonafork/horchata"></ref><ref name="bake-street/horchata"></ref> According to researchers at the University of Ilorin, made from tiger nuts is an inexpensive source of protein.


title: Horchata

text: The Valencian or is made with dried and sweetened tiger nuts ("Cyperus esculentus"). This form of horchata is now properly called .


title: Horchata

text: It remains popular in Spain, where a regulating council exists to ensure the quality and traceability of the product in relation to the designation of origin. There it is served ice-cold as a natural refreshment in the summer, often served with fartons.


title: Horchata

text: The majority of the Spanish tiger nut crop is utilised in the production of . Alboraya is the most important production centre. In total, sixteen Valencian municipalities grow tiger nuts protected by the quality seal, the only one that guarantees its Valencian origin, which in aggregate cover around 485 hectares of tiger nut fields, with an approximate annual production of 8,000 tonnes, which is normally all sold by the time it is harvested.


title: Horchata

text: It is the most common variety of in Mexico and Guatemala. In the United States, it is popular in and Mexican ice cream shops.


title: Horchata

text: Though was once typically homemade, it is now available in both ready-to-drink (shelf-stable or refrigerated) and powdered form in grocery stores, principally in the U.S. and Latin America.


title: Horchata

text: In the Central American countries of El Salvador, Nicaragua, Honduras, and Costa Rica, refers to the drink known as . Its base is made from grinding jicaro seeds, also locally referred to as "morro" seeds, with rice. Depending on the region, other additions include ground cocoa, cinnamon, sesame seeds, nutmeg, tiger nuts, vanilla, ground peanuts, almonds and cashews.


title: Horchata

text: In Nicaragua, it is made with semilla de jicaro and rice as a base. They are toasted and then ground into a fine powder. The powder is then mixed with water or milk and mixed with cinnamon powder and sugar. Cocoa beans are sometimes added to the horchata, also toasted and ground with the base.


title: Horchata

text: In Ecuador, is a clear red infusion or tea of 18 herbs, and is most famous in the province of Loja. Some of the herbs used are escancel or bloodleaf, lemon verbena, lemon grass, mint, chamomile, lemon balm, rose geranium, among others. It is not at all related to horchata of other Latin American countries, it simply shares the same denomination.


title: Horchata

text: The urban and rural people who consume the “horchata” drink in the Loja province report wellbeing benefits, and believe that this herbal mixture infusion promotes a healthy digestion, improves memory, and acts as an hepatic anti-inflammatory and a diuretic. There is also genotoxic activity in some of its ingredients.


title: Horchata

text: Horchata, as a flavor, makes appearances in ice cream, cookies, and other sweets, and other products such as RumChata, an alcoholic tribute to the beverage. Some smoothie shops, cafés, and McDonald's in the U.S. have been experimenting with horchata-flavored frappes.


title: Hilbert space

text: In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space.


title: Hilbert space

text: The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term "Hilbert space" for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.


title: Hilbert space

text: Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a linear subspace or a subspace (the analog of "dropping the altitude" of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis, in analogy with Cartesian coordinates in classical geometry. When this basis is countably infinite, it allows identifying the Hilbert space with the space of the infinite sequences that are square-summable. The latter space is often in the older literature referred to as "the" Hilbert space.


title: Hilbert space

text: One of the most familiar examples of a Hilbert space is the Euclidean vector space consisting of three-dimensional vectors, denoted by , and equipped with the dot product. The dot product takes two vectors and , and produces a real number . If and are represented in Cartesian coordinates, then the dot product is defined by


title: Hilbert space

text: An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted , and to the angle between two vectors and by means of the formula


title: Hilbert space

text: Multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist. A mathematical series


title: Hilbert space

text: consisting of vectors in is absolutely convergent provided that the sum of the lengths converges as an ordinary series of real numbers:


title: Hilbert space

text: Just as with a series of scalars, a series of vectors that converges absolutely also converges to some limit vector in the Euclidean space, in the sense that


title: Hilbert space

text: This property expresses the "completeness" of Euclidean space: that a series that converges absolutely also converges in the ordinary sense.


title: Hilbert space

text: Hilbert spaces are often taken over the complex numbers. The complex plane denoted by is equipped with a notion of magnitude, the complex modulus , which is defined as the square root of the product of with its complex conjugate:


title: Hilbert space

text: If is a decomposition of into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length:


title: Hilbert space

text: A second example is the space whose elements are pairs of complex numbers . Then the inner product of with another such vector is given by


title: Hilbert space

text: The real part of is then the two-dimensional Euclidean dot product. This inner product is "Hermitian" symmetric, which means that the result of interchanging and is the complex conjugate:


title: Hilbert space

text: A is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product.


title: Hilbert space

text: To say that is a means that is a complex vector space on which there is an inner product formula_11 associating a complex number to each pair of elements formula_12 of that satisfies the following properties:


title: Hilbert space

text: It follows from properties 1 and 2 that a complex inner product is , also called , in its second argument, meaning that


title: Hilbert space

text: A is defined in the same way, except that is a real vector space and the inner product takes real values. Such an inner product will be a bilinear map and formula_20 will form a dual system.


title: Hilbert space

text: That this function is a distance function means firstly that it is symmetric in formula_25 and formula_26 secondly that the distance between formula_25 and itself is zero, and otherwise the distance between formula_25 and formula_29 must be positive, and lastly that the triangle inequality holds, meaning that the length of one leg of a triangle cannot exceed the sum of the lengths of the other two legs:


title: Hilbert space

text: This last property is ultimately a consequence of the more fundamental Cauchy–Schwarz inequality, which asserts


title: Hilbert space

text: With a distance function defined in this way, any inner product space is a metric space, and sometimes is known as a . Any pre-Hilbert space that is additionally also a complete space is a Hilbert space.


title: Hilbert space

text: The of is expressed using a form of the Cauchy criterion for sequences in : a pre-Hilbert space is complete if every Cauchy sequence converges with respect to this norm to an element in the space. Completeness can be characterized by the following equivalent condition: if a series of vectors


title: Hilbert space

text: As a complete normed space, Hilbert spaces are by definition also Banach spaces. As such they are topological vector spaces, in which topological notions like the openness and closedness of subsets are well defined. Of special importance is the notion of a closed linear subspace of a Hilbert space that, with the inner product induced by restriction, is also complete (being a closed set in a complete metric space) and therefore a Hilbert space in its own right.


title: Hilbert space

text: The sequence space consists of all infinite sequences of complex numbers such that the following series converges:


title: Hilbert space

text: This second series converges as a consequence of the Cauchy–Schwarz inequality and the convergence of the previous series.


title: Hilbert space

text: Completeness of the space holds provided that whenever a series of elements from converges absolutely (in norm), then it converges to an element of . The proof is basic in mathematical analysis, and permits mathematical series of elements of the space to be manipulated with the same ease as series of complex numbers (or vectors in a finite-dimensional Euclidean space).


title: Hilbert space

text: Prior to the development of Hilbert spaces, other generalizations of Euclidean spaces were known to mathematicians and physicists. In particular, the idea of an abstract linear space (vector space) had gained some traction towards the end of the 19th century: this is a space whose elements can be added together and multiplied by scalars (such as real or complex numbers) without necessarily identifying these elements with "geometric" vectors, such as position and momentum vectors in physical systems. Other objects studied by mathematicians at the turn of the 20th century, in particular spaces of sequences (including series) and spaces of functions, can naturally be thought of as linear spaces. Functions, for instance, can be added together or multiplied by constant scalars, and these operations obey the algebraic laws satisfied by addition and scalar multiplication of spatial vectors.


title: Hilbert space

text: In the first decade of the 20th century, parallel developments led to the introduction of Hilbert spaces. The first of these was the observation, which arose during David Hilbert and Erhard Schmidt's study of integral equations, that two square-integrable real-valued functions and on an interval have an "inner product"


title: Hilbert space

text: which has many of the familiar properties of the Euclidean dot product. In particular, the idea of an orthogonal family of functions has meaning. Schmidt exploited the similarity of this inner product with the usual dot product to prove an analog of the spectral decomposition for an operator of the form


title: Hilbert space

text: where is a continuous function symmetric in and . The resulting eigenfunction expansion expresses the function as a series of the form


title: Hilbert space

text: where the functions are orthogonal in the sense that for all . The individual terms in this series are sometimes referred to as elementary product solutions. However, there are eigenfunction expansions that fail to converge in a suitable sense to a square-integrable function: the missing ingredient, which ensures convergence, is completeness.


title: Hilbert space

text: The second development was the Lebesgue integral, an alternative to the Riemann integral introduced by Henri Lebesgue in 1904. The Lebesgue integral made it possible to integrate a much broader class of functions. In 1907, Frigyes Riesz and Ernst Sigismund Fischer independently proved that the space of square Lebesgue-integrable functions is a complete metric space. As a consequence of the interplay between geometry and completeness, the 19th century results of Joseph Fourier, Friedrich Bessel and Marc-Antoine Parseval on trigonometric series easily carried over to these more general spaces, resulting in a geometrical and analytical apparatus now usually known as the Riesz–Fischer theorem.


title: Hilbert space

text: Further basic results were proved in the early 20th century. For example, the Riesz representation theorem was independently established by Maurice Fréchet and Frigyes Riesz in 1907. John von Neumann coined the term "abstract Hilbert space" in his work on unbounded Hermitian operators. Although other mathematicians such as Hermann Weyl and Norbert Wiener had already studied particular Hilbert spaces in great detail, often from a physically motivated point of view, von Neumann gave the first complete and axiomatic treatment of them. Von Neumann later used them in his seminal work on the foundations of quantum mechanics, and in his continued work with Eugene Wigner. The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups.


title: Hilbert space

text: The significance of the concept of a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics. In short, the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetries of the system are unitary operators, and measurements are orthogonal projections. The relation between quantum mechanical symmetries and unitary operators provided an impetus for the development of the unitary representation theory of groups, initiated in the 1928 work of Hermann Weyl. On the other hand, in the early 1930s it became clear that classical mechanics can be described in terms of Hilbert space (Koopman–von Neumann classical mechanics) and that certain properties of classical dynamical systems can be analyzed using Hilbert space techniques in the framework of ergodic theory.


title: Hilbert space

text: The algebra of observables in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg's matrix mechanics formulation of quantum theory. Von Neumann began investigating operator algebras in the 1930s, as rings of operators on a Hilbert space. The kind of algebras studied by von Neumann and his contemporaries are now known as von Neumann algebras. In the 1940s, Israel Gelfand, Mark Naimark and Irving Segal gave a definition of a kind of operator algebras called C*-algebras that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras. These techniques are now basic in abstract harmonic analysis and representation theory.


title: Hilbert space

text: Lebesgue spaces are function spaces associated to measure spaces , where is a set, is a σ-algebra of subsets of , and is a countably additive measure on . Let be the space of those complex-valued measurable functions on for which the Lebesgue integral of the square of the absolute value of the function is finite, i.e., for a function in ,


title: Hilbert space

text: where the second form (conjugation of the first element) is commonly found in the theoretical physics literature. For and in , the integral exists because of the Cauchy–Schwarz inequality, and defines an inner product on the space. Equipped with this inner product, is in fact complete. The Lebesgue integral is essential to ensure completeness: on domains of real numbers, for instance, not enough functions are Riemann integrable.


title: Hilbert space

text: The Lebesgue spaces appear in many natural settings. The spaces and of square-integrable functions with respect to the Lebesgue measure on the real line and unit interval, respectively, are natural domains on which to define the Fourier transform and Fourier series. In other situations, the measure may be something other than the ordinary Lebesgue measure on the real line. For instance, if is any positive measurable function, the space of all measurable functions on the interval satisfying


title: Hilbert space

text: The weighted space is identical with the Hilbert space where the measure of a Lebesgue-measurable set is defined by


title: Hilbert space

text: Weighted spaces like this are frequently used to study orthogonal polynomials, because different families of orthogonal polynomials are orthogonal with respect to different weighting functions.


title: Hilbert space

text: Sobolev spaces, denoted by or , are Hilbert spaces. These are a special kind of function space in which differentiation may be performed, but that (unlike other Banach spaces such as the Hölder spaces) support the structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenient setting for the theory of partial differential equations. They also form the basis of the theory of direct methods in the calculus of variations.


title: Hilbert space

text: For a non-negative integer and , the Sobolev space contains functions whose weak derivatives of order up to are also . The inner product in is


title: Hilbert space

text: where the dot indicates the dot product in the Euclidean space of partial derivatives of each order. Sobolev spaces can also be defined when is not an integer.


title: Hilbert space

text: Sobolev spaces are also studied from the point of view of spectral theory, relying more specifically on the Hilbert space structure. If is a suitable domain, then one can define the Sobolev space as the space of Bessel potentials; roughly,


title: Hilbert space

text: Here is the Laplacian and is understood in terms of the spectral mapping theorem. Apart from providing a workable definition of Sobolev spaces for non-integer , this definition also has particularly desirable properties under the Fourier transform that make it ideal for the study of pseudodifferential operators. Using these methods on a compact Riemannian manifold, one can obtain for instance the Hodge decomposition, which is the basis of Hodge theory.


title: Hilbert space

text: The Hardy spaces are function spaces, arising in complex analysis and harmonic analysis, whose elements are certain holomorphic functions in a complex domain. Let denote the unit disc in the complex plane. Then the Hardy space is defined as the space of holomorphic functions on such that the means


title: Hilbert space

text: Thus consists of those functions that are "L" on the circle, and whose negative frequency Fourier coefficients vanish.


title: Hilbert space

text: The Bergman spaces are another family of Hilbert spaces of holomorphic functions. Let be a bounded open set in the complex plane (or a higher-dimensional complex space) and let be the space of holomorphic functions in that are also in in the sense that


title: Hilbert space

text: where the integral is taken with respect to the Lebesgue measure in . Clearly is a subspace of ; in fact, it is a closed subspace, and so a Hilbert space in its own right. This is a consequence of the estimate, valid on compact subsets of , that


title: Hilbert space

text: which in turn follows from Cauchy's integral formula. Thus convergence of a sequence of holomorphic functions in implies also compact convergence, and so the limit function is also holomorphic. Another consequence of this inequality is that the linear functional that evaluates a function at a point of is actually continuous on . The Riesz representation theorem implies that the evaluation functional can be represented as an element of . Thus, for every , there is a function such that


title: Hilbert space

text: A Bergman space is an example of a reproducing kernel Hilbert space, which is a Hilbert space of functions along with a kernel that verifies a reproducing property analogous to this one. The Hardy space also admits a reproducing kernel, known as the Szegő kernel. Reproducing kernels are common in other areas of mathematics as well. For instance, in harmonic analysis the Poisson kernel is a reproducing kernel for the Hilbert space of square-integrable harmonic functions in the unit ball. That the latter is a Hilbert space at all is a consequence of the mean value theorem for harmonic functions.


title: Hilbert space

text: Many of the applications of Hilbert spaces exploit the fact that Hilbert spaces support generalizations of simple geometric concepts like projection and change of basis from their usual finite dimensional setting. In particular, the spectral theory of continuous self-adjoint linear operators on a Hilbert space generalizes the usual spectral decomposition of a matrix, and this often plays a major role in applications of the theory to other areas of mathematics and physics.


title: Hilbert space

text: In the theory of ordinary differential equations, spectral methods on a suitable Hilbert space are used to study the behavior of eigenvalues and eigenfunctions of differential equations. For example, the Sturm–Liouville problem arises in the study of the harmonics of waves in a violin string or a drum, and is a central problem in ordinary differential equations. The problem is a differential equation of the form


title: Hilbert space

text: The functions , , and are given in advance, and the problem is to find the function and constants for which the equation has a solution. The problem only has solutions for certain values of , called eigenvalues of the system, and this is a consequence of the spectral theorem for compact operators applied to the integral operator defined by the Green's function for the system. Furthermore, another consequence of this general result is that the eigenvalues of the system can be arranged in an increasing sequence tending to infinity.


title: Hilbert space

text: Hilbert spaces form a basic tool in the study of partial differential equations. For many classes of partial differential equations, such as linear elliptic equations, it is possible to consider a generalized solution (known as a weak solution) by enlarging the class of functions. Many weak formulations involve the class of Sobolev functions, which is a Hilbert space. A suitable weak formulation reduces to a geometrical problem the analytic problem of finding a solution or, often what is more important, showing that a solution exists and is unique for given boundary data. For linear elliptic equations, one geometrical result that ensures unique solvability for a large class of problems is the Lax–Milgram theorem. This strategy forms the rudiment of the Galerkin method (a finite element method) for numerical solution of partial differential equations.


title: Hilbert space

text: A typical example is the Poisson equation with Dirichlet boundary conditions in a bounded domain in . The weak formulation consists of finding a function such that, for all continuously differentiable functions in vanishing on the boundary:


title: Hilbert space

text: This can be recast in terms of the Hilbert space consisting of functions such that , along with its weak partial derivatives, are square integrable on , and vanish on the boundary. The question then reduces to finding in this space such that for all in this space


title: Hilbert space

text: Since the Poisson equation is elliptic, it follows from Poincaré's inequality that the bilinear form is coercive. The Lax–Milgram theorem then ensures the existence and uniqueness of solutions of this equation.


title: Hilbert space

text: Hilbert spaces allow for many elliptic partial differential equations to be formulated in a similar way, and the Lax–Milgram theorem is then a basic tool in their analysis. With suitable modifications, similar techniques can be applied to parabolic partial differential equations and certain hyperbolic partial differential equations.


title: Hilbert space

text: The field of ergodic theory is the study of the long-term behavior of chaotic dynamical systems. The protypical case of a field that ergodic theory applies to is thermodynamics, in which—though the microscopic state of a system is extremely complicated (it is impossible to understand the ensemble of individual collisions between particles of matter)—the average behavior over sufficiently long time intervals is tractable. The laws of thermodynamics are assertions about such average behavior. In particular, one formulation of the zeroth law of thermodynamics asserts that over sufficiently long timescales, the only functionally independent measurement that one can make of a thermodynamic system in equilibrium is its total energy, in the form of temperature.


title: Hilbert space

text: An ergodic dynamical system is one for which, apart from the energy—measured by the Hamiltonian—there are no other functionally independent conserved quantities on the phase space. More explicitly, suppose that the energy is fixed, and let be the subset of the phase space consisting of all states of energy (an energy surface), and let denote the evolution operator on the phase space. The dynamical system is ergodic if there are no continuous non-constant functions on such that


title: Hilbert space

text: for all on and all time . Liouville's theorem implies that there exists a measure on the energy surface that is invariant under the time translation. As a result, time translation is a unitary transformation of the Hilbert space consisting of square-integrable functions on the energy surface with respect to the inner product


title: Hilbert space

text: For an ergodic system, the fixed set of the time evolution consists only of the constant functions, so the ergodic theorem implies the following: for any function ,


title: Hilbert space

text: That is, the long time average of an observable is equal to its expectation value over an energy surface.


title: Hilbert space

text: One of the basic goals of Fourier analysis is to decompose a function into a (possibly infinite) linear combination of given basis functions: the associated Fourier series. The classical Fourier series associated to a function defined on the interval is a series of the form


title: Hilbert space

text: The example of adding up the first few terms in a Fourier series for a sawtooth function is shown in the figure. The basis functions are sine waves with wavelengths (for integer ) shorter than the wavelength of the sawtooth itself (except for , the "fundamental" wave). All basis functions have nodes at the nodes of the sawtooth, but all but the fundamental have additional nodes. The oscillation of the summed terms about the sawtooth is called the Gibbs phenomenon.


title: Hilbert space

text: A significant problem in classical Fourier series asks in what sense the Fourier series converges, if at all, to the function . Hilbert space methods provide one possible answer to this question. The functions form an orthogonal basis of the Hilbert space . Consequently, any square-integrable function can be expressed as a series


title: Hilbert space

text: The problem can also be studied from the abstract point of view: every Hilbert space has an orthonormal basis, and every element of the Hilbert space can be written in a unique way as a sum of multiples of these basis elements. The coefficients appearing on these basis elements are sometimes known abstractly as the Fourier coefficients of the element of the space. The abstraction is especially useful when it is more natural to use different basis functions for a space such as . In many circumstances, it is desirable not to decompose a function into trigonometric functions, but rather into orthogonal polynomials or wavelets for instance, and in higher dimensions into spherical harmonics.


title: Hilbert space

text: For instance, if are any orthonormal basis functions of , then a given function in can be approximated as a finite linear combination


title: Hilbert space

text: The coefficients are selected to make the magnitude of the difference as small as possible. Geometrically, the best approximation is the orthogonal projection of onto the subspace consisting of all linear combinations of the , and can be calculated by


title: Hilbert space

text: That this formula minimizes the difference is a consequence of Bessel's inequality and Parseval's formula.


title: Hilbert space

text: In various applications to physical problems, a function can be decomposed into physically meaningful eigenfunctions of a differential operator (typically the Laplace operator): this forms the foundation for the spectral study of functions, in reference to the spectrum of the differential operator. A concrete physical application involves the problem of hearing the shape of a drum: given the fundamental modes of vibration that a drumhead is capable of producing, can one infer the shape of the drum itself? The mathematical formulation of this question involves the Dirichlet eigenvalues of the Laplace equation in the plane, that represent the fundamental modes of vibration in direct analogy with the integers that represent the fundamental modes of vibration of the violin string.


title: Hilbert space

text: Spectral theory also underlies certain aspects of the Fourier transform of a function. Whereas Fourier analysis decomposes a function defined on a compact set into the discrete spectrum of the Laplacian (which corresponds to the vibrations of a violin string or drum), the Fourier transform of a function is the decomposition of a function defined on all of Euclidean space into its components in the continuous spectrum of the Laplacian. The Fourier transformation is also geometrical, in a sense made precise by the Plancherel theorem, that asserts that it is an isometry of one Hilbert space (the "time domain") with another (the "frequency domain"). This isometry property of the Fourier transformation is a recurring theme in abstract harmonic analysis (since it reflects the conservation of energy for the continuous Fourier Transform), as evidenced for instance by the Plancherel theorem for spherical functions occurring in noncommutative harmonic analysis.


title: Hilbert space

text: In the mathematically rigorous formulation of quantum mechanics, developed by John von Neumann, the possible states (more precisely, the pure states) of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a complex separable Hilbert space, known as the state space, well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projectivization of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system; for example, the position and momentum states for a single non-relativistic spin zero particle is the space of all square-integrable functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of spinors. Each observable is represented by a self-adjoint linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate.


title: Hilbert space

text: The inner product between two state vectors is a complex number known as a probability amplitude. During an ideal measurement of a quantum mechanical system, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states. The possible results of a measurement are the eigenvalues of the operator—which explains the choice of self-adjoint operators, for all the eigenvalues must be real. The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator.


title: Hilbert space

text: For a general system, states are typically not pure, but instead are represented as statistical mixtures of pure states, or mixed states, given by density matrices: self-adjoint operators of trace one on a Hilbert space. Moreover, for general quantum mechanical systems, the effects of a single measurement can influence other parts of a system in a manner that is described instead by a positive operator valued measure. Thus the structure both of the states and observables in the general theory is considerably more complicated than the idealization for pure states.


title: Hilbert space

text: Any true physical color can be represented by a combination of pure spectral colors. As physical colors can be composed of any number of spectral colors, the space of physical colors may aptly be represented by a Hilbert space over spectral colors. Humans have three types of cone cells for color perception, so the perceivable colors can be represented by 3-dimensional Euclidean space. The many-to-one linear mapping from the Hilbert space of physical colors to the Euclidean space of human perceivable colors explains why many distinct physical colors may be perceived by humans to be identical (e.g., pure yellow light versus a mix of red and green light, see metamerism).


title: Hilbert space

text: Two vectors and in a Hilbert space are orthogonal when . The notation for this is . More generally, when is a subset in , the notation means that is orthogonal to every element from .


title: Hilbert space

text: Whereas the Pythagorean identity as stated is valid in any inner product space, completeness is required for the extension of the Pythagorean identity to series. A series of "orthogonal" vectors converges in if and only if the series of squares of norms converges, and


title: Hilbert space

text: Furthermore, the sum of a series of orthogonal vectors is independent of the order in which it is taken.


title: Hilbert space

text: By definition, every Hilbert space is also a Banach space. Furthermore, in every Hilbert space the following parallelogram identity holds:


title: Hilbert space

text: Conversely, every Banach space in which the parallelogram identity holds is a Hilbert space, and the inner product is uniquely determined by the norm by the polarization identity. For real Hilbert spaces, the polarization identity is


title: Hilbert space

text: This subsection employs the Hilbert projection theorem. If is a non-empty closed convex subset of a Hilbert space and a point in , there exists a unique point that minimizes the distance between and points in ,


title: Hilbert space

text: This is equivalent to saying that there is a point with minimal norm in the translated convex set . The proof consists in showing that every minimizing sequence is Cauchy (using the parallelogram identity) hence converges (using completeness) to a point in that has minimal norm. More generally, this holds in any uniformly convex Banach space.


title: Hilbert space

text: When this result is applied to a closed subspace of , it can be shown that the point closest to is characterized by


title: Hilbert space

text: This point is the "orthogonal projection" of onto , and the mapping is linear (see Orthogonal complements and projections). This result is especially significant in applied mathematics, especially numerical analysis, where it forms the basis of least squares methods.


title: Hilbert space

text: In particular, when is not equal to , one can find a nonzero vector orthogonal to (select and ). A very useful criterion is obtained by applying this observation to the closed subspace generated by a subset of .


title: Hilbert space

text: The dual space is the space of all continuous linear functions from the space into the base field. It carries a natural norm, defined by


title: Hilbert space

text: This norm satisfies the parallelogram law, and so the dual space is also an inner product space where this inner product can be defined in terms of this dual norm by using the polarization identity. The dual space is also complete so it is a Hilbert space in its own right.


title: Hilbert space

text: If is a complete orthonormal basis for then the inner product on the dual space of any two formula_81 is


title: Hilbert space

text: The Riesz representation theorem affords a convenient description of the dual space. To every element of , there is a unique element of , defined by


title: Hilbert space

text: The Riesz representation theorem states that the map from to defined by is surjective, which makes this map an isometric antilinear isomorphism. So to every element of the dual there exists one and only one in such that


title: Hilbert space

text: The reversal of order on the right-hand side restores linearity in from the antilinearity of . In the real case, the antilinear isomorphism from to its dual is actually an isomorphism, and so real Hilbert spaces are naturally isomorphic to their own duals.


title: Hilbert space

text: The representing vector is obtained in the following way. When , the kernel is a closed vector subspace of , not equal to , hence there exists a nonzero vector orthogonal to . The vector is a suitable scalar multiple of . The requirement that yields


title: Hilbert space

text: This correspondence is exploited by the bra–ket notation popular in physics. It is common in physics to assume that the inner product, denoted by , is linear on the right,


title: Hilbert space

text: The Riesz representation theorem relies fundamentally not just on the presence of an inner product, but also on the completeness of the space. In fact, the theorem implies that the topological dual of any inner product space can be identified with its completion. An immediate consequence of the Riesz representation theorem is also that a Hilbert space is reflexive, meaning that the natural map from into its double dual space is an isomorphism.


title: Hilbert space

text: For example, any orthonormal sequence converges weakly to 0, as a consequence of Bessel's inequality. Every weakly convergent sequence is bounded, by the uniform boundedness principle.


title: Hilbert space

text: Conversely, every bounded sequence in a Hilbert space admits weakly convergent subsequences (Alaoglu's theorem). This fact may be used to prove minimization results for continuous convex functionals, in the same way that the Bolzano–Weierstrass theorem is used for continuous functions on . Among several variants, one simple statement is as follows:


title: Hilbert space

text: This fact (and its various generalizations) are fundamental for direct methods in the calculus of variations. Minimization results for convex functionals are also a direct consequence of the slightly more abstract fact that closed bounded convex subsets in a Hilbert space are weakly compact, since is reflexive. The existence of weakly convergent subsequences is a special case of the Eberlein–Šmulian theorem.


title: Hilbert space

text: Any general property of Banach spaces continues to hold for Hilbert spaces. The open mapping theorem states that a continuous surjective linear transformation from one Banach space to another is an open mapping meaning that it sends open sets to open sets. A corollary is the bounded inverse theorem, that a continuous and bijective linear function from one Banach space to another is an isomorphism (that is, a continuous linear map whose inverse is also continuous). This theorem is considerably simpler to prove in the case of Hilbert spaces than in general Banach spaces. The open mapping theorem is equivalent to the closed graph theorem, which asserts that a linear function from one Banach space to another is continuous if and only if its graph is a closed set. In the case of Hilbert spaces, this is basic in the study of unbounded operators (see closed operator).


title: Hilbert space

text: The (geometrical) Hahn–Banach theorem asserts that a closed convex set can be separated from any point outside it by means of a hyperplane of the Hilbert space. This is an immediate consequence of the best approximation property: if is the element of a closed convex set closest to , then the separating hyperplane is the plane perpendicular to the segment passing through its midpoint.


title: Hilbert space

text: The continuous linear operators from a Hilbert space to a second Hilbert space are "bounded" in the sense that they map bounded sets to bounded sets. Conversely, if an operator is bounded, then it is continuous. The space of such bounded linear operators has a norm, the operator norm given by


title: Hilbert space

text: The sum and the composite of two bounded linear operators is again bounded and linear. For "y" in "H", the map that sends to is linear and continuous, and according to the Riesz representation theorem can therefore be represented in the form


title: Hilbert space

text: for some vector in . This defines another bounded linear operator , the adjoint of . The adjoint satisfies . When the Riesz representation theorem is used to identify each Hilbert space with its continuous dual space, the adjoint of can be shown to be identical to the transpose of , which by definition sends formula_92 to the functional formula_93


title: Hilbert space

text: The set of all bounded linear operators on (meaning operators ), together with the addition and composition operations, the norm and the adjoint operation, is a C*-algebra, which is a type of operator algebra.


title: Hilbert space

text: An element of is called 'self-adjoint' or 'Hermitian' if . If is Hermitian and for every , then is called 'nonnegative', written ; if equality holds only when , then is called 'positive'. The set of self adjoint operators admits a partial order, in which if . If has the form for some , then is nonnegative; if is invertible, then is positive. A converse is also true in the sense that, for a non-negative operator , there exists a unique non-negative square root such that


title: Hilbert space

text: In a sense made precise by the spectral theorem, self-adjoint operators can usefully be thought of as operators that are "real". An element of is called "normal" if . Normal operators decompose into the sum of a self-adjoint operator and an imaginary multiple of a self adjoint operator


title: Hilbert space

text: that commute with each other. Normal operators can also usefully be thought of in terms of their real and imaginary parts.


title: Hilbert space

text: An element of is called unitary if is invertible and its inverse is given by . This can also be expressed by requiring that be onto and for all . The unitary operators form a group under composition, which is the isometry group of .


title: Hilbert space

text: An element of is compact if it sends bounded sets to relatively compact sets. Equivalently, a bounded operator is compact if, for any bounded sequence , the sequence has a convergent subsequence. Many integral operators are compact, and in fact define a special class of operators known as Hilbert–Schmidt operators that are especially important in the study of integral equations. Fredholm operators differ from a compact operator by a multiple of the identity, and are equivalently characterized as operators with a finite dimensional kernel and cokernel. The index of a Fredholm operator is defined by


title: Hilbert space

text: The index is homotopy invariant, and plays a deep role in differential geometry via the Atiyah–Singer index theorem.


title: Hilbert space

text: Unbounded operators are also tractable in Hilbert spaces, and have important applications to quantum mechanics. An unbounded operator on a Hilbert space is defined as a linear operator whose domain is a linear subspace of . Often the domain is a dense subspace of , in which case is known as a densely defined operator.


title: Hilbert space

text: The adjoint of a densely defined unbounded operator is defined in essentially the same manner as for bounded operators. Self-adjoint unbounded operators play the role of the "observables" in the mathematical formulation of quantum mechanics. Examples of self-adjoint unbounded operators on the Hilbert space are:


title: Hilbert space

text: These correspond to the momentum and position observables, respectively. Note that neither nor is defined on all of , since in the case of the derivative need not exist, and in the case of the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of .


title: Hilbert space

text: Two Hilbert spaces and can be combined into another Hilbert space, called the (orthogonal) direct sum, and denoted


title: Hilbert space

text: Each of the is included as a closed subspace in the direct sum of all of the . Moreover, the are pairwise orthogonal. Conversely, if there is a system of closed subspaces, , , in a Hilbert space , that are pairwise orthogonal and whose union is dense in , then is canonically isomorphic to the direct sum of . In this case, is called the internal direct sum of the . A direct sum (internal or external) is also equipped with a family of orthogonal projections onto the th direct summand . These projections are bounded, self-adjoint, idempotent operators that satisfy the orthogonality condition


title: Hilbert space

text: The spectral theorem for compact self-adjoint operators on a Hilbert space states that splits into an orthogonal direct sum of the eigenspaces of an operator, and also gives an explicit decomposition of the operator as a sum of projections onto the eigenspaces. The direct sum of Hilbert spaces also appears in quantum mechanics as the Fock space of a system containing a variable number of particles, where each Hilbert space in the direct sum corresponds to an additional degree of freedom for the quantum mechanical system. In representation theory, the Peter–Weyl theorem guarantees that any unitary representation of a compact group on a Hilbert space splits as the direct sum of finite-dimensional representations.


title: Hilbert space

text: If and , then one defines an inner product on the (ordinary) tensor product as follows. On simple tensors, let


title: Hilbert space

text: This formula then extends by sesquilinearity to an inner product on . The Hilbertian tensor product of and , sometimes denoted by , is the Hilbert space obtained by completing for the metric associated to this inner product.


title: Hilbert space

text: An example is provided by the Hilbert space . The Hilbertian tensor product of two copies of is isometrically and linearly isomorphic to the space of square-integrable functions on the square . This isomorphism sends a simple tensor to the function


title: Hilbert space

text: This example is typical in the following sense. Associated to every simple tensor product is the rank one operator from to that maps a given as


title: Hilbert space

text: This mapping defined on simple tensors extends to a linear identification between and the space of finite rank operators from to . This extends to a linear isometry of the Hilbertian tensor product with the Hilbert space of Hilbert–Schmidt operators from to .


title: Hilbert space

text: The notion of an orthonormal basis from linear algebra generalizes over to the case of Hilbert spaces. In a Hilbert space , an orthonormal basis is a family of elements of satisfying the conditions:


title: Hilbert space

text: A system of vectors satisfying the first two conditions basis is called an orthonormal system or an orthonormal set (or an orthonormal sequence if is countable). Such a system is always linearly independent.


title: Hilbert space

text: Despite the name, an orthonormal basis is not, in general, a basis in the sense of linear algebra (Hamel basis). More precisely, an orthonormal basis is a Hamel basis if and only if the Hilbert space is a finite-dimensional vector space.


title: Hilbert space

text: This is related to the fact that the only vector orthogonal to a dense linear subspace is the zero vector, for if is any orthonormal set and is orthogonal to , then is orthogonal to the closure of the linear span of , which is the whole space.


title: Hilbert space

text: In the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter basis is also called a Hamel basis. That the span of the basis vectors is dense implies that every vector in the space can be written as the sum of an infinite series, and the orthogonality implies that this decomposition is unique.


title: Hilbert space

text: The space formula_109 of square-summable sequences of complex numbers is the set of infinite sequences


title: Hilbert space

text: This space is the infinite-dimensional generalization of the formula_113 space of finite-dimensional vectors. It is usually the first example used to show that in infinite-dimensional spaces, a set that is closed and bounded is not necessarily (sequentially) compact (as is the case in all "finite" dimensional spaces). Indeed, the set of orthonormal vectors above shows this: It is an infinite sequence of vectors in the unit ball (i.e., the ball of points with norm less than or equal one). This set is clearly bounded and closed; yet, no subsequence of these vectors converges to anything and consequently the unit ball in formula_109 is not compact. Intuitively, this is because "there is always another coordinate direction" into which the next elements of the sequence can evade.


title: Hilbert space

text: One can generalize the space formula_109 in many ways. For example, if is any (infinite) set, then one can form a Hilbert space of sequences with index set , defined by


title: Hilbert space

text: the supremum being taken over all finite subsets of . It follows that, for this sum to be finite, every element of has only countably many nonzero terms. This space becomes a Hilbert space with the inner product


title: Hilbert space

text: for all . Here the sum also has only countably many nonzero terms, and is unconditionally convergent by the Cauchy–Schwarz inequality.


title: Hilbert space

text: Then for every . It follows that is orthogonal to each , hence is orthogonal to . Using the Pythagorean identity twice, it follows that


title: Hilbert space

text: Let , be an arbitrary orthonormal system in . Applying the preceding inequality to every finite subset of gives Bessel's inequality:


title: Hilbert space

text: Geometrically, Bessel's inequality implies that the orthogonal projection of onto the linear subspace spanned by the has norm that does not exceed that of . In two dimensions, this is the assertion that the length of the leg of a right triangle may not exceed the length of the hypotenuse.


title: Hilbert space

text: Bessel's inequality is a stepping stone to the stronger result called Parseval's identity, which governs the case when Bessel's inequality is actually an equality. By definition, if is an orthonormal basis of , then every element of may be written as


title: Hilbert space

text: Even if is uncountable, Bessel's inequality guarantees that the expression is well-defined and consists only of countably many nonzero terms. This sum is called the Fourier expansion of , and the individual coefficients are the Fourier coefficients of . Parseval's identity then asserts that


title: Hilbert space

text: Conversely, if is an orthonormal set such that Parseval's identity holds for every , then is an orthonormal basis.


title: Hilbert space

text: As a consequence of Zorn's lemma, "every" Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality, called the Hilbert dimension of the space. For instance, since has an orthonormal basis indexed by , its Hilbert dimension is the cardinality of (which may be a finite integer, or a countable or uncountable cardinal number).


title: Hilbert space

text: The Hilbert dimension is not greater than the Hamel dimension (the usual dimension of a vector space). The two dimensions are equal if and only one is finite.


title: Hilbert space

text: As a consequence of Parseval's identity, if is an orthonormal basis of , then the map defined by is an isometric isomorphism of Hilbert spaces: it is a bijective linear mapping such that


title: Hilbert space

text: for all . The cardinal number of is the Hilbert dimension of . Thus every Hilbert space is isometrically isomorphic to a sequence space for some set .


title: Hilbert space

text: By definition, a Hilbert space is separable provided it contains a dense countable subset. Along with Zorn's lemma, this means a Hilbert space is separable if and only if it admits a countable orthonormal basis. All infinite-dimensional separable Hilbert spaces are therefore isometrically isomorphic to .


title: Hilbert space

text: In the past, Hilbert spaces were often required to be separable as part of the definition. Most spaces used in physics are separable, and since these are all isomorphic to each other, one often refers to any infinite-dimensional separable Hilbert space as ""the" Hilbert space" or just "Hilbert space". Even in quantum field theory, most of the Hilbert spaces are in fact separable, as stipulated by the Wightman axioms. However, it is sometimes argued that non-separable Hilbert spaces are also important in quantum field theory, roughly because the systems in the theory possess an infinite number of degrees of freedom and any infinite Hilbert tensor product (of spaces of dimension greater than one) is non-separable. For instance, a bosonic field can be naturally thought of as an element of a tensor product whose factors represent harmonic oscillators at each point of space. From this perspective, the natural state space of a boson might seem to be a non-separable space. However, it is only a small separable subspace of the full tensor product that can contain physically meaningful fields (on which the observables can be defined). Another non-separable Hilbert space models the state of an infinite collection of particles in an unbounded region of space. An orthonormal basis of the space is indexed by the density of the particles, a continuous parameter, and since the set of possible densities is uncountable, the basis is not countable.


title: Hilbert space

text: The set is a closed subspace of (can be proved easily using the linearity and continuity of the inner product) and so forms itself a Hilbert space. If is a closed subspace of , then is called the of . In fact, every can then be written uniquely as , with and . Therefore, is the internal Hilbert direct sum of and .


title: Hilbert space

text: The linear operator that maps to is called the onto . There is a natural one-to-one correspondence between the set of all closed subspaces of and the set of all bounded self-adjoint operators such that . Specifically,


title: Hilbert space

text: This provides the geometrical interpretation of : it is the best approximation to "x" by elements of "V".


title: Hilbert space

text: Projections and are called mutually orthogonal if . This is equivalent to and being orthogonal as subspaces of . The sum of the two projections and is a projection only if and are orthogonal to each other, and in that case . The composite is generally not a projection; in fact, the composite is a projection if and only if the two projections commute, and in that case .


title: Hilbert space

text: By restricting the codomain to the Hilbert space , the orthogonal projection gives rise to a projection mapping ; it is the adjoint of the inclusion mapping


title: Hilbert space

text: Every closed subspace "V" of a Hilbert space is therefore the image of an operator of norm one such that . The property of possessing appropriate projection operators characterizes Hilbert spaces:


title: Hilbert space

text: While this result characterizes the metric structure of a Hilbert space, the structure of a Hilbert space as a topological vector space can itself be characterized in terms of the presence of complementary subspaces:


title: Hilbert space

text: The orthogonal complement satisfies some more elementary results. It is a monotone function in the sense that if , then with equality holding if and only if is contained in the closure of . This result is a special case of the Hahn–Banach theorem. The closure of a subspace can be completely characterized in terms of the orthogonal complement: if is a subspace of , then the closure of is equal to . The orthogonal complement is thus a Galois connection on the partial order of subspaces of a Hilbert space. In general, the orthogonal complement of a sum of subspaces is the intersection of the orthogonal complements:


title: Hilbert space

text: There is a well-developed spectral theory for self-adjoint operators in a Hilbert space, that is roughly analogous to the study of symmetric matrices over the reals or self-adjoint matrices over the complex numbers. In the same sense, one can obtain a "diagonalization" of a self-adjoint operator as a suitable sum (actually an integral) of orthogonal projection operators.


title: Hilbert space

text: The spectrum of an operator , denoted , is the set of complex numbers such that lacks a continuous inverse. If is bounded, then the spectrum is always a compact set in the complex plane, and lies inside the disc . If is self-adjoint, then the spectrum is real. In fact, it is contained in the interval where


title: Hilbert space

text: Unlike with finite matrices, not every element of the spectrum of must be an eigenvalue: the linear operator may only lack an inverse because it is not surjective. Elements of the spectrum of an operator in the general sense are known as "spectral values". Since spectral values need not be eigenvalues, the spectral decomposition is often more subtle than in finite dimensions.


title: Hilbert space

text: However, the spectral theorem of a self-adjoint operator takes a particularly simple form if, in addition, is assumed to be a compact operator. The spectral theorem for compact self-adjoint operators states:


title: Hilbert space

text: This theorem plays a fundamental role in the theory of integral equations, as many integral operators are compact, in particular those that arise from Hilbert–Schmidt operators.


title: Hilbert space

text: The general spectral theorem for self-adjoint operators involves a kind of operator-valued Riemann–Stieltjes integral, rather than an infinite summation. The "spectral family" associated to associates to each real number λ an operator , which is the projection onto the nullspace of the operator , where the positive part of a self-adjoint operator is defined by


title: Hilbert space

text: The operators are monotone increasing relative to the partial order defined on self-adjoint operators; the eigenvalues correspond precisely to the jump discontinuities. One has the spectral theorem, which asserts


title: Hilbert space

text: The integral is understood as a Riemann–Stieltjes integral, convergent with respect to the norm on . In particular, one has the ordinary scalar-valued integral representation


title: Hilbert space

text: A somewhat similar spectral decomposition holds for normal operators, although because the spectrum may now contain non-real complex numbers, the operator-valued Stieltjes measure must instead be replaced by a resolution of the identity.


title: Hilbert space

text: A major application of spectral methods is the spectral mapping theorem, which allows one to apply to a self-adjoint operator any continuous complex function defined on the spectrum of by forming the integral


title: Hilbert space

text: The resulting continuous functional calculus has applications in particular to pseudodifferential operators.


title: Hilbert space

text: The spectral theory of "unbounded" self-adjoint operators is only marginally more difficult than for bounded operators. The spectrum of an unbounded operator is defined in precisely the same way as for bounded operators: is a spectral value if the resolvent operator


title: Hilbert space

text: fails to be a well-defined continuous operator. The self-adjointness of still guarantees that the spectrum is real. Thus the essential idea of working with unbounded operators is to look instead at the resolvent where is nonreal. This is a "bounded" normal operator, which admits a spectral representation that can then be transferred to a spectral representation of itself. A similar strategy is used, for instance, to study the spectrum of the Laplace operator: rather than address the operator directly, one instead looks as an associated resolvent such as a Riesz potential or Bessel potential.


title: Hilbert space

text: In "Gravity's Rainbow" (1973), a novel by Thomas Pynchon, one of the characters is called "Sammy Hilbert-Spaess", a pun on "Hilbert Space". The novel refers also to Gödel's incompleteness theorems.


title: D'arcy Wretzky

text: D'arcy Elizabeth Wretzky-Brown (born May 1, 1968) is an American musician. She is the original bass player of the alternative rock band the Smashing Pumpkins and is credited on their first six studio albums. She left the band in 1999.


title: D'arcy Wretzky

text: Wretzky was born and raised in South Haven, Michigan, where her mother, Vikke Anderson, a working musician, encouraged D'arcy and her sisters to perform music. Growing up, she played the violin and oboe and performed in choirs. She also performed gymnastics. Wretzky intended to start a musical career since she was 10 years old. She would later refer to her father, Jerry Wretzky, a pipefitter with a love of horseback riding, as "a very strange man". The young Wretzky was a self-described "tomboy", and had a contentious relationship with her sister.


title: D'arcy Wretzky

text: She attended South Haven's L.C. Mohr High School, where she grew interested in post-punk and played in cover bands. After high school, she moved to France to join a band, but the band had already disbanded upon her arrival, prompting her to return to the United States. She then moved to Chicago and spent the summer living with friends and attending concerts.


title: D'arcy Wretzky

text: Wretzky's legal name, "Wretzky-Brown", comes from her having been married to musician Kerry Brown, from 1993 to 1999. According to Wretzky, she had a miscarriage in 1999.


title: D'arcy Wretzky

text: After a concert at a local rock club, Wretzky overheard Billy Corgan criticizing the band that had performed. An argument and discussion followed, and Corgan recruited her into his band, the nascent Smashing Pumpkins, which, at the time, was merely Corgan, James Iha, and a drum machine. Wretzky accepted, and Jimmy Chamberlin completed the lineup a few months later, after Joe Shanahan encouraged Corgan to add a live drummer.


title: D'arcy Wretzky

text: Wretzky is the credited bassist on the Smashing Pumpkins' first five studio albums: "Gish", "Siamese Dream", "Mellon Collie and the Infinite Sadness", "Adore", and "Machina/The Machines of God". It was confirmed by both her and Corgan, however, that Corgan played the bass tracks on "Gish" and "Siamese Dream" because he could complete them in far fewer takes. Wretzky often contributed backing vocals in concert, and on studio albums. She contributes vocally in some Smashing Pumpkins songs including "Daydream" from "Gish", many songs on "Siamese Dream", "1979", "Cupid De Locke", "Farewell and Goodnight", "Beautiful"; "Where Boys Fear to Tread" from "Mellon Collie", and "Dreaming" and "The Bells" from "The Aeroplane Flies High". Wretzky also co-wrote one Smashing Pumpkins song, "Daughter".


title: D'arcy Wretzky

text: In 1995, Wretzky and Iha started an independent record label called Scratchie Records, featuring artists such as The Frogs.


title: D'arcy Wretzky

text: Wretzky's time in the band was marked by alternating periods of happiness and discomfort. Corgan considered her the "moral authority" and "moral conscience" of the band. In the aftermath of the success of 1995's "Mellon Collie and the Infinite Sadness", Corgan said she began an "apparent slow descent into insanity and/or drugs (take your pick)." After the short, nine-date "The Arising!" tour in April 1999 which saw all four original members performing together for the first time since 1996, Wretzky decided to leave the band with intentions of pursuing an acting career. The band was recording "Machina/The Machines of God" and "Machina II/The Friends and Enemies of Modern Music" at the time and consequently she performed very few bass parts on the album. Most of the bass parts were handled by Corgan himself. Shortly after leaving the group, she was arrested for possession of crack cocaine. Corgan later said she was "fired for being a mean-spirited drug addict who refused to get help." She was replaced on 2000's "Machina" tours by former Hole bassist Melissa Auf der Maur.


title: D'arcy Wretzky

text: Wretzky did not participate in the Smashing Pumpkins' reunion. In 2008, she and her former boyfriend and bandmate James Iha filed a lawsuit against Virgin Records for selling ringtones of Smashing Pumpkins songs without their consent.


title: D'arcy Wretzky

text: After many years out of the spotlight, Wretzky resurfaced in July 2009 by calling in unexpectedly on Chicago's Q101 FM with Ryan Manno. During the interview, she stated that she was not healthy enough to be a musician, and repeatedly professed her admiration for Monkees frontman Davy Jones who was known to be an early romantic crush of Wretzky's. She also discussed her appreciation for the band Silversun Pickups who have a sound influenced by the early "Gish" era of the Smashing Pumpkins. She also mentioned that she then lived on a farm in Michigan, that she had briefly lived in Austin, Texas, sometime during the previous decade, and that her former fiancé Wendell Green had died.


title: D'arcy Wretzky

text: Wretzky was jailed on February 1, 2011, for missing four court dates related to a ticket she received for failing to control her wild horses, allowing them to freely roam the streets at night causing interference to local traffic and farmers, as well as trespassing on public property and stealing vegetables from the local farmer's market storage. She spent six days in jail. The following day after getting released from jail, she was arrested again on February 7, 2011, on a misdemeanor drunken-driving road rage charge in South Haven, Michigan. She was sent back to jail.


title: D'arcy Wretzky

text: In 2013, Wretzky's relationship with Corgan remained tense, with both parties saying that they were not speaking. In September 2014, she tried to re-establish contact with him, but Corgan had changed his phone number.


title: D'arcy Wretzky

text: In August 2016, Corgan posted a video to Facebook stating that he and Wretzky had recently reestablished communication saying "I've been in communication with D'arcy for the first time in 16 or 17 years, it's awesome to have my friend back." Corgan emphasized that this did not necessarily mean the band was getting back together, instead insisting "my primary interest in the old band was us having good relationships again."


title: D'arcy Wretzky

text: On February 14, 2018, Wretzky gave her first interview in nearly 20 years. In it, she was highly critical of Corgan and their past.


title: D'arcy Wretzky

text: Wretzky stated she had been offered a contract to rejoin the band on the Shiny and Oh So Bright tour, but that Corgan rescinded the offer soon after. Corgan released a statement denying the claims, stating "Ms. Wretzky has repeatedly been invited out to play with the group, participate in demo sessions, or at the very least, meet face-to-face, and in each and every instance she always deferred". Wretzky later provided screenshots of a text message conversation with Corgan that appeared to support her version of the story.


title: D'arcy Wretzky

text: Wretzky joined the band Catherine as a second vocalist for their final album, "Hot Saki and Bedtime Stories." She also appeared in the video for "Four Leaf Clover." At the time, Wretzky was married to Catherine member Kerry Brown.


title: D'arcy Wretzky

text: Wretzky contributed vocals to the track "One and Two" on James Iha's 1998 solo album, "Let It Come Down."


title: D'arcy Wretzky

text: In 1999, she worked with cellist Eric Remschneider, whom she had worked with when he had recorded with the Smashing Pumpkins. That year, she also contributed backing vocals on the Filter song "Cancer" from "Title of Record."


title: D'arcy Wretzky

text: In May 2016 in an interview with "Loudwire," Filter lead singer Richard Patrick spoke of a romantic relationship he had with Wretzky, saying she was the subject of a song he wrote called "Miss Blue", also on "Title of Record."


title: D'arcy Wretzky

text: While making no indication that she will record with or make any meaningful contribution, Michigan doom-sludge metal band Grave Next Door said on Twitter that they were jamming with D'arcy in her home, with one band member posting a pic of himself with a prop from the video for "Tonight Tonight".


title: Coaxial cable

text: Coaxial cable, or coax (pronounced ) is a type of electrical cable consisting of an inner conductor surrounded by a concentric conducting shield, with the two separated by a dielectric (insulating material); many coaxial cables also have a protective outer sheath or jacket. The term "coaxial" refers to the inner conductor and the outer shield sharing a geometric axis.


title: Coaxial cable

text: Coaxial cable is a type of transmission line, used to carry high-frequency electrical signals with low losses. It is used in such applications as telephone trunk lines, broadband internet networking cables, high-speed computer data busses, cable television signals, and connecting radio transmitters and receivers to their antennas. It differs from other shielded cables because the dimensions of the cable and connectors are controlled to give a precise, constant conductor spacing, which is needed for it to function efficiently as a transmission line.


title: Coaxial cable

text: Coaxial cable was used in the first (1858) and following transatlantic cable installations, but its theory was not described until 1880 by English physicist, engineer, and mathematician Oliver Heaviside, who patented the design in that year (British patent No. 1,407).


title: Coaxial cable

text: Coaxial cable is used as a transmission line for radio frequency signals. Its applications include feedlines connecting radio transmitters and receivers to their antennas, computer network (e.g., Ethernet) connections, digital audio (S/PDIF), and distribution of cable television signals. One advantage of coaxial over other types of radio transmission line is that in an ideal coaxial cable the electromagnetic field carrying the signal exists only in the space between the inner and outer conductors. This allows coaxial cable runs to be installed next to metal objects such as gutters without the power losses that occur in other types of transmission lines. Coaxial cable also provides protection of the signal from external electromagnetic interference.