Topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space (an algebraic structure) which is also a topological space, this implies that vector space operations are continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence. The elements of topological vector spaces are typically functions or linear operators acting on topological vector spaces, and the topology is often defined so as to capture  a particular notion of convergence of sequences of functions. Banach spaces, Hilbert spaces and Sobolev spaces are well-known examples. Unless stated otherwise, the underlying field of a topological vector space is assumed to be either the complex numbers formula_1 or the real numbers formula_2 Motivation. Normed spaces Every normed vector space has a natural topological structure: the norm induces a metric and the metric induces a topology. This is a topological vector space because: Thus all Banach spaces and Hilbert spaces are examples of topological vector spaces. Non-normed spaces There are topological vector spaces whose topology is not  induced by a norm, but are still of interest in analysis. Examples of such spaces are spaces of holomorphic functions on an open domain, spaces of infinitely differentiable functions, the Schwartz spaces, and spaces of test functions and the spaces of distributions on them. These are all examples of Montel spaces. An infinite-dimensional Montel space is never normable. The existence of a norm for a given topological vector space is characterized by Kolmogorov's normability criterion. A topological field is a topological vector space over each of its subfields. Definition. A topological vector space (TVS) formula_7 is a vector  space over a topological field formula_5 (most often the real or complex numbers with their standard topologies) that is endowed with a topology such that vector addition formula_3 and scalar multiplication formula_10 are continuous functions (where the domains of these functions are endowed with product topologies). Such a topology is called a ' or a ' on formula_11 Every topological vector space is also a commutative topological group under addition. Hausdorff assumption Some authors (for example, Walter Rudin) require the topology on formula_7 to be T; it then follows that the space is Hausdorff, and even Tychonoff. A topological vector space is  said to be if it is Hausdorff; importantly, "separated" does not mean separable. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed below. Category and morphisms The category of topological vector spaces over a given topological field formula_5 is commonly denoted TVS or TVect. The objects are the topological vector spaces over formula_5 and the morphisms are the continuous formula_5-linear maps from one object to another. A ' (abbreviated ') or topological homomorphism is a continuous linear map formula_16 between topological vector spaces (TVSs) such that the  induced map formula_17 is an open mapping when formula_18 which is the range or image of formula_19 is given the subspace topology induced by "Y". A ' (abbreviated ') or a topological monomorphism is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding. A ' (abbreviated '), also called a ' or an ', is a bijective linear homeomorphism. Equivalently, it is a surjective TVS embedding Many properties of TVSs that are studied, such as local convexity, metrizability, completeness, and normability, are invariant under TVS isomorphisms. A necessary condition for a vector topology A collection formula_20 of  subsets of a vector space is called additive if for every formula_21 there exists some formula_22 such that formula_23 All of the above conditions are consequently a necessity for a topology to form a vector topology. Defining topologies using neighborhoods of the origin. Since every vector topology is translation invariant (which means that for all formula_24 the map formula_25 defined by formula_26 is a homeomorphism), to define a vector topology it suffices to define a neighborhood basis (or subbasis) for it at the origin. In general, the set of all balanced and absorbing subsets of a vector space does not  satisfy the conditions of this theorem and does not form a neighborhood basis at the origin for any vector topology. Defining topologies using strings. Let formula_7 be a vector space and let formula_28 be a sequence of subsets of formula_11 Each set in the sequence formula_30 is called a of formula_30 and for every index formula_32 formula_33 is called the formula_34 knot of formula_35 The set formula_36 is called the beginning of formula_35 The sequence formula_30 is/is a: If formula_46is an absorbing disk in a vector space formula_7 then the sequence defined by formula_48 forms a string beginning with  formula_49 This is called the natural string of formula_46 Moreover, if a vector space formula_7 has countable dimension then every string contains an absolutely convex string. Summative sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions. These functions can then be used to prove many of the basic properties of topological vector spaces. A proof of the above theorem is given in the article on metrizable TVSs. If formula_52 and formula_53 are two collections of subsets of a vector space formula_7 and if formula_55 is a scalar, then by definition: If  formula_66 is a collection sequences of subsets of formula_6 then formula_66 is said to be directed (downwards) under inclusion or simply directed if formula_66 is not empty and for all formula_70 there exists some formula_71 such that formula_72 and formula_73 (said differently, if and only if formula_66 is a prefilter with respect to the containment formula_75 defined above). Notation: Let formula_76 be the set of all knots of all strings in formula_77 Defining vector topologies using collections of strings is particularly useful for defining classes of TVSs that are not necessarily locally convex. If formula_66 is the set of all  topological strings in a TVS formula_79 then formula_80 A Hausdorff TVS is metrizable if and only if its topology can be induced by a single topological string. Topological structure. A vector space is an abelian group with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by −1). Hence, every topological vector space is an abelian topological group. Every TVS is completely regular but a TVS need not be normal. Let formula_7 be a topological vector space. Given a subspace formula_82 the quotient space  formula_83 with the usual quotient topology is a Hausdorff topological vector space if and only if formula_84 is closed. This permits the following construction: given a topological vector space formula_7 (that is probably not Hausdorff), form the quotient space formula_83 where formula_84 is the closure of formula_88 formula_83 is then a Hausdorff topological vector space that can be studied instead of formula_11 Invariance of vector topologies. One of the most used properties of vector topologies is that every vector topology is translation invariant: Scalar multiplication by a non-zero scalar is a TVS-isomorphism. This means that if formula_95 then  the linear map formula_25 defined by formula_97 is a homeomorphism. Using formula_98 produces the negation map formula_25 defined by formula_100 which is consequently a linear homeomorphism and thus a TVS-isomorphism. If formula_101 and any subset formula_102 then formula_103 and moreover, if formula_104 then formula_105 is a neighborhood (resp. open neighborhood, closed neighborhood) of formula_106 in formula_7 if and only if the same is true of formula_108 at the origin. Local notions. A subset formula_109 of a vector space formula_7 is said to be Every neighborhood of 0 is an absorbing set and contains an open balanced neighborhood of formula_124  so every topological vector space has a local base of absorbing and balanced sets. The origin even has a neighborhood basis consisting of closed balanced neighborhoods of 0; if the space is locally convex then it also has a neighborhood basis consisting of closed convex balanced neighborhoods of 0. Bounded subsets A subset formula_109 of a topological vector space formula_7 is bounded if for every neighborhood formula_127 of the origin, then formula_128 when formula_129 is sufficiently large. The definition of boundedness can be weakened a bit; formula_109 is bounded if and only if every countable subset of it is bounded . A set is bounded if and only if each of its subsequences is a bounded set. Also, formula_109 is bounded if and only if for every balanced neighborhood formula_127 of 0, there exists formula_129 such that formula_134 Moreover, when formula_7 is locally convex, the boundedness can be characterized by seminorms: the subset formula_109 is bounded if and only if every continuous seminorm formula_137 is bounded on formula_138 Every totally bounded set is bounded. If formula_84 is a vector subspace of a TVS formula_6 then a subset of formula_84 is bounded in formula_84 if and only if it is bounded in  formula_11 Metrizability. A TVS is pseudometrizable if and only if it has a countable neighborhood basis at the origin, or equivalent, if and only if its topology is generated by an "F"-seminorm. A TVS is metrizable if and only if it is Hausdorff and pseudometrizable. More strongly: a topological vector space is said to be normable if its topology can be induced by a norm. A topological vector space is normable if and only if it is Hausdorff and has a convex bounded neighborhood of formula_144 Let formula_5 be a non-discrete locally compact topological field, for example the real  or complex numbers. A Hausdorff topological vector space over formula_5 is locally compact if and only if it is finite-dimensional, that is, isomorphic to formula_147 for some natural number formula_148 Completeness and uniform structure. The canonical uniformity on a TVS formula_79 is the unique translation-invariant uniformity that induces the topology formula_150 on formula_11 Every TVS is assumed to be endowed with this canonical uniformity, which makes all TVSs into uniform spaces. This allows one to about related notions such as completeness, uniform convergence, Cauchy nets, and uniform continuity. etc., which are always assumed to be with respect to  this uniformity (unless indicated other). This implies that every Hausdorff topological vector space is Tychonoff. A subspace of a TVS is compact if and only if it is complete and totally bounded (for Hausdorff TVSs, a set being totally bounded is equivalent to it being precompact). But if the TVS is not Hausdorff then there exist compact subsets that are not closed. However, the closure of a compact subset of a non-Hausdorff TVS is again compact (so compact subsets are relatively compact). With respect to this uniformity, a net (or sequence) formula_152 is Cauchy if and only if for  every neighborhood formula_127 of formula_154 there exists some index formula_155 such that formula_156 whenever formula_157 and formula_158 Every Cauchy sequence is bounded, although Cauchy nets and Cauchy filters may not be bounded. A topological vector space where every Cauchy sequence converges is called sequentially complete; in general, it may not be complete (in the sense that all Cauchy filters converge). The vector space operation of addition is uniformly continuous and an open map. Scalar multiplication is Cauchy continuous but in general, it is almost never uniformly continuous. Because of this, every topological vector space can be completed and is thus  a dense linear subspace of a complete topological vector space. Examples. Finest and coarsest vector topology. Let formula_7 be a real or complex vector space. Trivial topology The trivial topology or indiscrete topology formula_171 is always a TVS topology on any vector space formula_7 and it is the coarsest TVS topology possible. An important consequence of this is that the intersection of any collection of TVS topologies on formula_7 always contains a TVS topology. Any vector space (including those that are infinite dimensional) endowed with the trivial topology is a compact (and thus locally compact) complete pseudometrizable seminormable locally convex  topological vector space. It is Hausdorff if and only if formula_174 Finest vector topology There exists a TVS topology formula_175 on formula_6 called the on formula_6 that is finer than every other TVS-topology on formula_7 (that is, any TVS-topology on formula_7 is necessarily a subset of formula_175). Every linear map from formula_181 into another TVS is necessarily continuous. If formula_7 has an uncountable Hamel basis then formula_175 is locally convex and metrizable. Product vector spaces. A Cartesian product of a family of topological vector spaces, when endowed with the product topology, is a topological vector space. Consider for  instance the set formula_7 of all functions formula_185 where formula_186 carries its usual Euclidean topology. This set formula_7 is a real vector space (where addition and scalar multiplication are defined pointwise, as usual) that can be identified with (and indeed, is often defined to be) the Cartesian product formula_188 which carries the natural product topology. With this product topology, formula_189 becomes a topological vector space whose topology is called . The reason for this name is the following: if formula_190 is a sequence (or more generally, a net) of elements in formula_7 and if formula_192 then formula_193 converges to formula_194 in  formula_7 if and only if for every real number formula_196 formula_197 converges to formula_198 in formula_2 This TVS is complete, Hausdorff, and locally convex but not metrizable and consequently not normable; indeed, every neighborhood of the origin in the product topology contains lines (that is, 1-dimensional vector subspaces, which are subsets of the form formula_200 with formula_201). Finite-dimensional spaces. By F. Riesz's theorem, a Hausdorff topological vector space is finite-dimensional if and only if it is locally compact, which happens if and only if it has a compact neighborhood of the origin. Let formula_5 denote formula_186  or formula_1 and endow formula_5 with its usual Hausdorff normed Euclidean topology. Let formula_7 be a vector space over formula_5 of finite dimension formula_208 and so that formula_7 is vector space isomorphic to formula_147 (explicitly, this means that there exists a linear isomorphism between the vector spaces formula_7 and formula_147). This finite-dimensional vector space formula_7 always has a unique vector topology, which makes it TVS-isomorphic to formula_214 where formula_147 is endowed with the usual Euclidean topology (which is the same as the product topology). This Hausdorff vector topology is also the (unique) finest vector topology on formula_11 formula_7  has a unique vector topology if and only if formula_174 If formula_219 then although formula_7 does not have a unique vector topology, it does have a unique vector topology. The proof of this dichotomy is straightforward so only an outline with the important observations is given. As usual, formula_5 is assumed have the (normed) Euclidean topology. Let formula_229 for all formula_230 Let formula_7 be a 1-dimensional vector space over formula_232 If formula_233 and formula_234 is a ball centered at 0 then formula_235 whenever formula_108 contains an "unbounded sequence", by which it is meant a sequence of the form formula_237  where formula_238 and formula_239 is unbounded in normed space formula_5 (in the usual sense). Any vector topology on formula_7 will be translation invariant and invariant under non-zero scalar multiplication, and for every formula_242 the map formula_243 given by formula_244 is a continuous linear bijection. Because formula_245 for any such formula_196 every subset of formula_7 can be written as formula_248 for some unique subset formula_249 And if this vector topology on formula_7 has a neighborhood formula_251 of the origin that is not equal to all of formula_6 then the continuity of scalar multiplication formula_253 at the origin guarantees the existence  of an open ball formula_254 centered at formula_124 and an open neighborhood formula_108 of the origin in formula_7 such that formula_258 which implies that formula_108 does contain any "unbounded sequence". This implies that for every formula_242 there exists some positive integer formula_155 such that formula_262 From this, it can be deduced that if formula_7 does not carry the trivial topology and if formula_242 then for any ball formula_234 center at 0 in formula_266 formula_267 contains an open neighborhood of the origin in formula_6 which then proves that formula_269 is a linear homeomorphism. formula_270 Non-vector topologies. Discrete and cofinite topologies  If formula_7 is a non-trivial vector space (that is, of non-zero dimension) then the discrete topology on formula_7 (which is always metrizable) is a TVS topology because despite making addition and negation continuous (which makes it into a topological group under addition), it fails to make scalar multiplication continuous. The cofinite topology on formula_7 (where a subset is open if and only if its complement is finite) is also a TVS topology on formula_11 Linear maps. A linear operator between two topological vector spaces which is continuous at one point is continuous on the whole domain. Moreover, a  linear operator formula_194 is continuous if formula_294 is bounded (as defined below) for some neighborhood formula_7 of the origin. A hyperplane on a topological vector space formula_7 is either dense or closed. A linear functional formula_194 on a topological vector space formula_7 has either dense or closed kernel. Moreover, formula_194 is continuous if and only if its kernel is closed. Types. Depending on the application additional constraints are usually enforced on the topological structure of the space. In fact, several principal results in functional analysis fail to hold in general for topological vector spaces: the closed graph theorem, the open  mapping theorem, and the fact that the dual space of the space separates points in the space. Below are some common topological vector spaces, roughly in order of increasing "niceness." Dual space. Every topological vector space has a continuous dual space—the set formula_320 of all continuous linear functionals, that is, continuous linear maps from the space into the base field formula_232 A topology on the dual can be defined to be the coarsest topology such that the dual pairing each point evaluation formula_322 is continuous. This turns the dual into a locally convex topological vector space. This topology is  called the weak-* topology. This may not be the only natural topology on the dual space; for instance, the dual of a normed space has a natural norm defined on it. However, it is very important in applications because of its compactness properties (see Banach–Alaoglu theorem). Caution: Whenever formula_7 is a non-normable locally convex space, then the pairing map formula_324 is never continuous, no matter which vector space topology one chooses on formula_325 A topological vector space has a non-trivial continuous dual space if and only if it has a proper convex neighborhood of the origin. Properties . For any formula_233 of a TVS formula_6 the "convex" (resp. "balanced, disked, closed convex, closed balanced, closed disked"') "hull" of formula_108 is the smallest subset of formula_7 that has this property and contains formula_330 The closure (respectively, interior, convex hull, balanced hull, disked hull) of a set formula_108 is sometimes denoted by formula_332 (respectively, formula_333). The convex hull formula_334 of a subset formula_108 is equal to the set of all of elements in formula_336 which are finite linear combinations of the form formula_337 where formula_338 is an integer, formula_339 and formula_340 sum to formula_341 The intersection of any family of  convex sets is convex and the convex hull of a subset is equal to the intersection of all convex sets that contain it. Neighborhoods and open sets. Properties of neighborhoods and open sets Every TVS is connected and locally connected and any connected open subset of a TVS is arcwise connected. If formula_233 and formula_46 is an open subset of formula_7 then formula_345 is an open set in formula_7 and if formula_233 has non-empty interior then formula_348 is a neighborhood of the origin. The open convex subsets of a TVS formula_7 (not necessarily Hausdorff or locally convex) are exactly  those that are of the form formula_350 for some formula_351 and some positive continuous sublinear functional formula_137 on formula_11 If formula_354 is an absorbing disk in a TVS formula_7 and if formula_356 is the Minkowski functional of formula_354 then formula_358 where importantly, it was assumed that formula_354 had any topological properties nor that formula_137 was continuous (which happens if and only if formula_354 is a neighborhood of 0). Let formula_150 and formula_363 be two vector topologies on formula_11 Then formula_365 if and only if whenever a net formula_152 in formula_7 converges formula_124 in formula_369 then formula_168 in formula_371 Let formula_20  be a neighborhood basis of the origin in formula_6 let formula_102 and let formula_375 Then formula_376 if and only if there exists a net formula_377 in formula_108 (indexed by formula_20) such that formula_380 in formula_11 This shows, in particular, that it will often suffice to consider nets indexed by a neighborhood basis of the origin rather than nets on arbitrary directed sets. If formula_7 is a TVS that is of the second category in itself (that is, a nonmeager space) then any closed convex absorbing subset of formula_7 is a neighborhood of the origin. This is no longer guaranteed if  the set is not convex (a counter-example exists even in formula_384) or if formula_7 is not of the second category in itself. Interior If formula_386 and formula_108 has non-empty interior then formula_388 and formula_389 If formula_108 is a disk in formula_7 that has non-empty interior then the origin belongs to the interior of formula_330 However, a closed balanced subset of formula_7 with non-empty interior may fail to contain the origin in its interior. If formula_108 is a balanced subset of formula_7 with non-empty interior then formula_396 is balanced; in particular, if the interior of a  balanced set contains the origin then formula_397 is balanced. If formula_159 is convex and formula_399 then formula_400 If formula_401 is any balanced neighborhood of the origin in formula_7 then formula_403 where formula_404 is the set of all scalars formula_405 such that formula_406 If formula_106 belongs to the interior of a convex set formula_233 and formula_409 then the half-open line segment formula_410 and formula_411 If formula_412 is a balanced neighborhood of formula_124 in formula_7 and formula_415 then by considering intersections of the form formula_416 (which are convex symmetric neighborhoods of formula_124 in the real TVS formula_418) it follows that: formula_419  and furthermore, if formula_420 then formula_421 and if formula_422 then formula_423 Non-Hausdorff spaces and the closure of the origin. A topological vector space formula_7 is Hausdorff if and only if formula_425 is a closed subset of formula_6 or equivalently, if and only if formula_427 Because formula_425 is a vector subspace of formula_6 the same is true of its closure formula_430 which is referred to as in formula_11 This vector space satisfies formula_432 so that in particular, every neighborhood of the origin in formula_7 contains the vector space formula_434 as a subset. The subspace topology on formula_434 is always the  trivial topology, which in particular implies that the topological vector space formula_434 a compact space (even if its dimension is non-zero or even infinite) and consequently also a bounded subset of formula_11 In fact, a vector subspace of a TVS is bounded if and only if it is contained in the closure of formula_88 Every subset of formula_434 also carries the trivial topology and so is itself a compact, and thus also complete, subspace (see footnote for a proof). In particular, if formula_7 is not Hausdorff then there exist subsets that are both but in formula_7; for instance, this  will be true of any non-empty proper subset of formula_442 If formula_233 is compact, then formula_444 and this set is compact. Thus the closure of a compact subset of a TVS is compact (said differently, all compact sets are relatively compact), which is not guaranteed for arbitrary non-Hausdorff topological spaces. For every subset formula_102 formula_446 and consequently, if formula_233 is open or closed in formula_7 then formula_449 (so that this open closed subsets formula_108 can be described as a "tube" whose vertical side is the vector space formula_434). For any subset formula_233 of this TVS formula_6 the following  are equivalent: If formula_84 is a vector subspace of a TVS formula_7 then formula_83 is Hausdorff if and only if formula_84 is closed in formula_11 Moreover, the quotient map formula_464 is always a closed map onto the (necessarily) Hausdorff TVS. Every vector subspace of formula_7 that is an algebraic complement of formula_434 (that is, a vector subspace formula_467 that satisfies formula_468 and formula_469) is a topological complement of formula_442 Consequently, if formula_467 is an algebraic complement of formula_434 in formula_7 then the addition map formula_474 defined by formula_475 is a TVS-isomorphism, where formula_467 is necessarily Hausdorff and formula_434 has  the indiscrete topology. Moreover, if formula_159 is a Hausdorff completion of formula_467 then formula_480 is a completion of formula_481 Closed and compact sets. Compact and totally bounded sets A subset of a TVS is compact if and only if it is complete and totally bounded. Thus, in a complete topological vector space, a closed and totally bounded subset is compact. A subset formula_108 of a TVS formula_7 is totally bounded if and only if formula_332 is totally bounded, if and only if its image under the canonical quotient map formula_485 is totally bounded. Every relatively compact set is totally bounded  and the closure of a totally bounded set is totally bounded. The image of a totally bounded set under a uniformly continuous map (such as a continuous linear map for instance) is totally bounded. If formula_108 is a subset of a TVS formula_7 such that every sequence in formula_108 has a cluster point in formula_108 then formula_108 is totally bounded. If formula_354 is a compact subset of a TVS formula_7 and formula_46 is an open subset of formula_7 containing formula_495 then there exists a neighborhood formula_412 of 0 such that formula_497 Closure and closed set The closure of any convex  (respectively, any balanced, any absorbing) subset of any TVS has this same property. In particular, the closure of any convex, balanced, and absorbing subset is a barrel. The closure of a vector subspace of a TVS is a vector subspace. Every finite dimensional vector subspace of a Hausdorff TVS is closed. The sum of a closed vector subspace and a finite-dimensional vector subspace is closed. If formula_84 is a vector subspace of formula_7 and formula_412 is a closed neighborhood of the origin in formula_7 such that formula_502 is closed in formula_7 then formula_84 is closed in formula_11 The sum  of a compact set and a closed set is closed. However, the sum of two closed subsets may fail to be closed (see this footnote for examples). If formula_233 and formula_405 is a scalar then formula_508 where if formula_7 is Hausdorff, formula_510 then equality holds: formula_511 In particular, every non-zero scalar multiple of a closed set is closed. If formula_233 and if formula_513 is a set of scalars such that neither formula_514 contain zero then formula_515 If formula_516 then formula_332 is convex. If formula_386 then formula_519 and so consequently, if formula_520 is closed then so is formula_521 If formula_7  is a real TVS and formula_102 then formula_524 where the left hand side is independent of the topology on formula_525 moreover, if formula_108 is a convex neighborhood of the origin then equality holds. For any subset formula_102 formula_528 where formula_20 is any neighborhood basis at the origin for formula_11 However, formula_531 and it is possible for this containment to be proper (for example, if formula_532 and formula_108 is the rational numbers). It follows that formula_534 for every neighborhood formula_46 of the origin in formula_11 Closed hulls In a locally convex space, convex hulls of bounded sets are bounded. This is  not true for TVSs in general. If formula_386 and the closed convex hull of one of the sets formula_108 or formula_542 is compact then formula_543 If formula_386 each have a closed convex hull that is compact (that is, formula_545 and formula_546 are compact) then formula_547 Hulls and compactness In a general TVS, the closed convex hull of a compact set may to be compact. The balanced hull of a compact (resp. totally bounded) set has that same property. The convex hull of a finite union of compact sets is again compact and convex. Other properties. Meager, nowhere dense, and Baire  A disk in a TVS is not nowhere dense if and only if its closure is a neighborhood of the origin. A vector subspace of a TVS that is closed but not open is nowhere dense. Suppose formula_7 is a TVS that does not carry the indiscrete topology. Then formula_7 is a Baire space if and only if formula_7 has no balanced absorbing nowhere dense subset. A TVS formula_7 is a Baire space if and only if formula_7 is nonmeager, which happens if and only if there does not exist a nowhere dense set formula_553 such that formula_554 Every nonmeager  locally convex TVS is a barrelled space. Important algebraic facts and common misconceptions If formula_233 then formula_556; if formula_108 is convex then equality holds. For an example where equality does hold, let formula_106 be non-zero and set formula_559 formula_560 also works. A subset formula_159 is convex if and only if formula_562 for all positive real formula_563 or equivalently, if and only if formula_564 for all formula_120 The convex balanced hull of a set formula_233 is equal to the convex hull of the balanced hull of formula_567 that is, it is equal to formula_568 But in general, formula_569 where the  inclusion might be strict since the balanced hull of a convex set need not be convex (counter-examples exist even in formula_570). If formula_386 and formula_405 is a scalar then formula_573 If formula_386 are convex non-empty disjoint sets and formula_575 then formula_576 In any non-trivial vector space formula_6 there exist two disjoint non-empty convex subsets whose union is formula_11 Other properties Every TVS topology can be generated by a of "F"-seminorms. If formula_579 is some unary predicate (a true or false statement dependent on formula_101) then for any formula_581 formula_582 So for example, if formula_579 denotes "formula_584  then for any formula_581 formula_586 Similarly, if formula_95 is a scalar then formula_588 The elements formula_101 of these sets must range over a vector space (that is, over formula_7) rather than not just a subset or else these equalities are no longer guaranteed; similarly, formula_591 must belong to this vector space (that is, formula_351). Properties preserved by set operators. The following table, the color of each cell indicates whether or not a given property of subsets of formula_7 (indicated by the column name, convex" for instance) is preserved under the set operator (indicated by the row's name, "closure" for  instance). If in every TVS, a property is preserved under the indicated set operator then that cell will be colored green; otherwise, it will be colored red. So for instance, since the union of two absorbing sets is again absorbing, the cell in row "formula_594" and column "Absorbing" is colored green. But since the arbitrary intersection of absorbing sets need not be absorbing, the cell in row "Arbitrary intersections (of at least 1 set)" and column "Absorbing" is colored red. If a cell is not colored then that information has yet to be filled in.