nyuuzyou/wikis huggingface dataset:
Additive Inverse in Ring of Continuous Mappings

== Theorem ==
Let $\struct {S, \tau_{_S} }$ be a topological space.
Let $\struct {R, +, *, \tau_{_R} }$ be a topological ring with zero $0_R$.
Let $\struct{\map C {S, R}, +, *}$ be the ring of continuous mappings from $S$ to $R$.
Let $f \in \map C {S, R}$.
Then:
:the additive inverse of $f$ is the pointwise negation $-f$ defined by:
::$\forall s \in S : \map {\paren{-f}} s = - \map f s$
== Proof ==
Let $\struct {R^S, +, *}$ be the ring of mappings from $S$ to $R$.
From Ring of Continuous Mappings is Subring of All Mappings:
:$\struct{\map C {S, R}, +, *}$ is a subring of $\struct {R^S, +, *}$
From Structure Induced by Ring Operations is Ring:
:$\forall f \in R^S :$ the additive inverse of $f$ is the pointwise negation $-f$, defined by:
::$\forall s \in S: \map {\paren {-f} } s := - \map f s$
The result follows.
== Also see ==
* Ring of Continuous Mappings is Subring of All Mappings
* Zero of Ring of Continuous Mappings
* Unity of Ring of Continuous Mappings
* Commutativity of Ring of Continuous Mappings
Category:Rings of Continuous Mappings
Additive Inverse in Ring of Continuous Real-Valued Functions

== Theorem ==
Let $\struct {S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $\struct {\map C {S, \R}, +, *}$ be the ring of continuous real-valued functions from $S$.
Let $f \in \map C {S, \R}$.
Then:
:the additive inverse of $f$ is the pointwise negation $-f$ defined by:
::$\forall s \in S : \map {-f} s = - \map f s$
== Proof ==
By definition of ring of continuous real-valued functions:
:$\struct {\map C {S, \R}, +, *}$ is the ring of continuous mappings from $S$ to $\R$.
From Additive Inverse in Ring of Continuous Mappings:
:$\forall f \in \map C {S, \R} :$ the additive inverse of $f$ is the pointwise negation $-f$, defined by:
::$\forall s \in S: \map {\paren {-f} } s := - \map f s$
The result follows.
== Also see ==
* Ring of Continuous Real-Valued Functions is Ring
* Zero of Ring of Continuous Real-Valued Functions
* Unity of Ring of Continuous Real-Valued Functions
* Ring of Continuous Real-Valued Functions is Commutative
== Sources ==
: Chapter $1$: Functions on a Topological Space, $\S 1.3$
Category:Rings of Continuous Real-Valued Functions
Additive Inverse in Vector Space is Unique

== Theorem ==
Let $\struct {\mathbf V, +, \circ}_F$ be a vector space over a field $F$, as defined by the vector space axioms.
Then for every $\mathbf v \in \mathbf V$, the additive inverse of $\mathbf v$ is unique:
:$\forall \mathbf v \in \mathbf V: \exists! \paren {-\mathbf v} \in \mathbf V: \mathbf v + \paren {-\mathbf v} = \mathbf 0$
== Proof ==
=== Proof of Existence ===
Follows from the vector space axioms.
=== Proof of Uniqueness ===
Let $\mathbf v$ have inverses $\mathbf x$ and $\mathbf y$.
Then:
== Also see ==
* Zero Vector is Unique
* Vector Inverse is Negative Vector
== Sources ==
* : Chapter $7$: Vector Spaces: $\S 32$. Definition of a Vector Space: Theorem $64 \ \text{(ii)}$
* : p. 15
Category:Vector Algebra
Category:Linear Algebra
Category:Vector Spaces
Additive Nowhere Negative Function is Subadditive

== Theorem ==
Let $\AA$ be an algebra of sets.
Let $f: \AA \to \overline \R$ be an additive function such that:
:$\forall A \in \AA: \map f A \ge 0$
Then $f$ is subadditive.
== Proof ==
If $f$ is additive then by Additive Function is Strongly Additive:
:$\forall A, B \in \AA: \map f {A \cup B} = \map f A + \map f B - \map f {A \cap B}$
As $\map f {A \cap B} \ge 0$, the result follows by definition of subadditive:
:$\forall A, B \in \AA: \map f {A \cup B} \le \map f A + \map f B$
Category:Set Systems
Additive Regular Representations of Topological Ring are Homeomorphisms

== Theorem ==
Let $\struct {R, + , \circ, \tau}$ be a topological ring.
Let $x \in R$.
Let $\lambda_x$ and $\rho_x$ be the left and right regular representations of $\struct {R, +}$ with respect to $x$.
Then $\lambda_x, \,\rho_x: \struct {R, \tau} \to \struct {R, \tau}$ are homeomorphisms with inverses $\lambda_{-x}, \,\rho_{-x}: \struct {R, \tau} \to \struct {R, \tau}$ respectively.
== Proof ==
By definition of a topological ring, $ \struct {R, + , \tau}$ is a topological group.
From Right and Left Regular Representations in Topological Group are Homeomorphisms:
:$\lambda_x, \,\rho_x: \struct {R, \tau} \to \struct {R, \tau}$ are homeomorphisms with inverses $\,\lambda_{-x}, \,\rho_{-x}: \struct {R, \tau} \to \struct {R, \tau}$ respectively.
Category:Topological Rings
Additive and Countably Subadditive Function is Countably Additive

== Theorem ==
Let $\Sigma$ be a $\sigma$-algebra over a set $X$.
Let $f: \Sigma \to \overline \R_{\ge 0}$ be an additive and countably subadditive function, where $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers.
Then $f$ is countably additive.
== Proof ==
Let $\sequence {S_n}_{n \mathop \in \N}$ be a sequence of pairwise disjoint elements of $\Sigma$.
Let $N \in \N$ be any natural number.
By Set is Subset of Union:
:$\ds \bigcup_{n \mathop = 0}^N S_n \subseteq \bigcup_{n \mathop = 0}^\infty S_n$
From Non-Negative Additive Function is Monotone:
:$\ds \map f {\bigcup_{n \mathop = 0}^N S_n} \le \map f {\bigcup_{n \mathop = 0}^\infty S_n}$
Also, from Finite Union of Sets in Additive Function:
:$\ds \map f {\bigcup_{n \mathop = 0}^N S_n} = \sum_{n \mathop = 0}^N \map f {S_n}$
Hence:
:$\ds \sum_{n \mathop = 0}^N \map f {S_n} \le \map f {\bigcup_{n \mathop = 0}^\infty S_n}$
Taking the limit as $N \to \infty$, it follows by Lower and Upper Bounds for Sequences that:
:$\ds \sum_{n \mathop = 0}^\infty \map f {S_n} \le \map f {\bigcup_{n \mathop = 0}^\infty S_n}$
The reverse inequality holds because of the countable subadditivity of $f$, and thus equality holds.
Category:Measure Theory
Additivity of Riemannian Length of Admissible Curve

== Theorem ==
Let $\struct {M, g}$ be a Riemannian manifold.
Let $\closedint a b$ be a closed real interval.
Let $\gamma : \closedint a b \to M$ with $t \stackrel \gamma \mapsto \map \gamma t$ be an admissible curve.
Let $c \in \R$ be a real number such that $a < c < b$.
Let $\map {L_g} {\valueat \gamma {\closedint a b} }$ be the Riemannian length of $\gamma$ from $t = a$ to $t = b$.
Then:
:$\map {L_g} {\valueat \gamma {\closedint a b} } = \map {L_g} {\valueat \gamma {\closedint a c} } + \map {L_g} {\valueat \gamma {\closedint c b} }$
== Proof ==
== Sources ==
* : $\S 2$: Riemannian Metrics. Lengths and Distances
Category:Curves
Category:Riemannian Geometry
Adjacency Matrix/Examples/Digraph

== Example of Adjacency Matrix ==
The adjacency matrix for a directed graph is no longer symmetrical about the main diagonal:
 $\qquad \begin{pmatrix}
0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 \\
\end{pmatrix}$
Category:Examples of Adjacency Matrices
Adjacency Matrix/Examples/Loop-Digraph

== Example of Adjacency Matrix ==
The adjacency matrix for a loop-digraph is not only not symmetrical about the main diagonal, it also has non-zero elements on that main diagonal:
 $\qquad \begin{pmatrix}
1 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 \\
\end{pmatrix}$
Category:Examples of Adjacency Matrices
Adjacency Matrix/Examples/Loop-Graph

== Example of Adjacency Matrix ==
The elements on the main diagonal of the adjacency matrix for a loop-graph are not all non-zero:
 $\qquad \begin{pmatrix}
0 & 1 & 0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 2 & 1 & 1 & 1 & 0 \\
0 & 0 & 1 & 2 & 1 & 1 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
\end{pmatrix}$
== Also defined as ==
Some other treatments of this subject require that a loop contributes a value of $1$ to the vertex to which it is incident.
Category:Examples of Adjacency Matrices
Adjacency Matrix/Examples/Loop-Multigraph

== Example of Adjacency Matrix ==
The elements of the adjacency matrix of a loop-multigraph are integers, and this time some of the diagonal elements are non-zero:
 $\qquad \begin{pmatrix}
0 & 2 & 0 & 0 & 0 & 0 \\
2 & 0 & 1 & 1 & 0 & 0 \\
0 & 1 & 2 & 1 & 0 & 0 \\
0 & 1 & 1 & 2 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 3 \\
0 & 0 & 0 & 1 & 3 & 0 \\
\end{pmatrix}$
Category:Examples of Adjacency Matrices
Adjacency Matrix/Examples/Multigraph

== Example of Adjacency Matrix ==
The elements of the adjacency matrix of a multigraph are integers, and the diagonal elements are all zero:
 $\qquad \begin{pmatrix}
0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 2 & 1 & 0 & 0 \\
0 & 2 & 0 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 0 & 3 \\
0 & 0 & 0 & 0 & 3 & 0 \\
\end{pmatrix}$
Category:Examples of Adjacency Matrices
Adjacency Matrix/Examples/Network

== Example of Adjacency Matrix ==
The adjacency matrix for a general network can have any numbers in any of its elements:
 $\qquad \begin{pmatrix}
0 & 1.2 & 0 & 0.5 \\
0 & 0 & 0 & 3.4 \\
0 & 0 & 0 & 0 \\
4.1 & 0 & 10.1 & 0 \\
\end{pmatrix}$
Category:Examples of Adjacency Matrices
Adjacency Matrix/Examples/Simple Graph

== Example of Adjacency Matrix ==
The elements of the adjacency matrix of a simple graph are $0$ and $1$, and the diagonal elements are all zero:
: $\qquad \begin{pmatrix}
0 & 1 & 1 & 0 \\
1 & 0 & 1 & 0 \\
1 & 1 & 0 & 1 \\
0 & 0 & 1 & 0 \\
\end{pmatrix}$
== Sources ==
* 
* 
* 
* 
Category:Examples of Adjacency Matrices
Adjoined Number/Examples/Root of 2

== Example of Adjoined Number ==
Consider the example of a field extension:
:Real Numbers of Type $a + b \sqrt 2: a, b \in \Q$
where $\Q \sqbrk {\sqrt 2}$ forms a finite field extension over the rational numbers $\Q$ of degree $2$.
The element $\sqrt 2$ is an example of an adjoined number on $\Q$.
== Sources ==
* 
* 
Category:Adjoined Numbers
Adjoining Commutes with Inverting

== Theorem ==
Let $\HH$ and $\KK$ be Hilbert spaces.
Let $\map \BB {\HH, \KK}$ be the set of bounded linear transformations from $\HH$ to $\KK$.
Let $A \in \map \BB {\HH, \KK}$ be a bounded linear transformation on $\HH$.
Let $A^{-1} \in \map \BB {\KK, \HH}$ be an inverse for $A$.
Let $A^*$ denote the adjoint of $A$.
Then $A^*$, is invertible, and:
:$\paren {A^*}^{-1} = \paren {A^{-1} }^*$
== Proof ==
By definition of inverse, one has $A A^{-1} = I_\KK$, where $I_\KK$ is the identity operator on $\KK$.
From Adjoint of Composition of Linear Transformations is Composition of Adjoints and Adjoint of Identity Transformation:
:$I_\KK = {I_\KK}^* = \paren {A A^{-1} }^* = \paren {A^{-1} }^*A^*$
Similarly:
:$I_\HH = {I_\HH}^* = \paren {A^{-1} A}^* = A^* \paren {A^{-1} }^*$
Hence, by definition of inverse:
:$\paren {A^*}^{-1} = \paren {A^{-1} }^*$
Hence, by definition, $A^*$ is invertible.
== Sources ==
* $\text {II}.2.6 \ \text {(d)}$
Category:Adjoints
Adjoining is Linear

== Theorem ==
Let $\HH$ and $\KK$ be Hilbert spaces over $\Bbb F \in \set {\R, \C}$.
Let $\map \BB {\HH, \KK}$ be the set of bounded linear transformations from $\HH$ to $\KK$.
Let $A, B \in \map \BB {\HH, \KK}$ be bounded linear transformations.
Then the operation of adjoining $^*$ satisfies, for all $\lambda \in \Bbb F$:
:$(1): \quad \paren {\lambda A}^* = \overline \lambda A^*$
:$(2): \quad \paren {A + B}^* = A^* + B^*$
where $\overline \lambda$ denotes the complex conjugate of $\lambda$.
That is,:
:$^*: \map \BB {\HH, \KK} \to \map \BB {\KK, \HH}$
is a linear transformation.
== Proof ==
Let $\innerprod \cdot \cdot_\HH$ and $\innerprod \cdot \cdot_\KK$ be inner products on $\HH$ and $\KK$ respectively.
=== Proof of $(1)$ ===
Let $\lambda \in \Bbb F$, $h \in \HH, k \in \KK$.
Then:
Thus, by Existence and Uniqueness of Adjoint:
:$\paren {\lambda A}^* = \overline \lambda A^*$
=== Proof of $(2)$ ===
Let $h \in \HH, k \in \KK$.
Then:
Thus, by Existence and Uniqueness of Adjoint:
:$\paren {A + B}^* = A^* + B^*$
== Sources ==
* $\text {II}.2.6 \ \text {(a)}$
* $13.1$: Existence of Hilbert Adjoint
Category:Adjoints
Adjoint is Involutive

== Theorem ==
Let $\mathbb F \in \set {\R, \C}$.
Let $\struct {\HH, \innerprod \cdot \cdot_\HH}$ and $\struct {\KK, \innerprod \cdot \cdot}$ be Hilbert spaces over $\mathbb F$.
Let $A : \HH \to \KK$ be a bounded linear transformation. 
Define:
:$A^{\ast \ast} = \paren {A^\ast}^\ast$
where $A^*$ denotes the adjoint of $A$. 
Then $\ast$ is an involutive operation in the sense that:
:$A^{\ast \ast} = A$
== Proof ==
Let $x \in \HH$ and $y \in \KK$. 
By the definition of the adjoint, we have: 
:$\innerprod {A y} x_\HH = \innerprod y {A^\ast x}_\KK$
Taking complex conjugates, we have: 
:$\overline {\innerprod {A y} x_\HH} = \overline {\innerprod y {A^\ast x}_\KK}$
Using the conjugate symmetry of the inner product, we have: 
:$\innerprod x {A y}_\HH = \innerprod {A^\ast x} y_\KK$
So by the definition of the adjoint:
:$\innerprod x {A y}_\HH = \innerprod x {\paren {A^\ast}^\ast y}_\KK$ 
From the uniqueness of the adjoint, we obtain: 
:$\paren {A^\ast}^\ast = A$
== Sources ==
* $\text {II}.2.6 \ \text {(c)}$
Category:Adjoints
Adjoint of Closure of Closable Densely-Defined Linear Operator

== Theorem ==
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\tuple {\map D T, T}$ be a closable densely-defined linear operator with adjoint $\tuple {\map D {T^\ast}, T^\ast}$. 
Let $\tuple {\map D {\overline T}, \overline T}$ be the closure of $T$ with adjoint $\tuple {\map D {\paren {\overline T}^\ast}, \paren {\overline T}^\ast}$. 
Then:
:$\tuple {\map D {T^\ast}, T^\ast} = \tuple {\map D {\paren {\overline T}^\ast}, \paren {\overline T}^\ast}$
== Proof ==
From Adjoint of Densely-Defined Linear Operator is Closed:
:$\tuple {\map D {T^\ast}, T^\ast}$ is closed.
From Closable Densely-Defined Operator is Closed iff Equal to Closure, we have:
:$\tuple {\map D {T^\ast}, T^\ast} = \tuple {\map D {\overline {T^\ast} }, \overline {T^\ast} }$
From Densely-Defined Linear Operator is Closable iff Adjoint is Densely-Defined, we have:
:$\tuple {\map D {\overline {T^\ast} }, T^\ast} = \tuple {\map D {T^{\ast \ast \ast} }, T^{\ast \ast \ast} }$
Writing $T^{\ast \ast \ast} = \paren {T^{\ast \ast} }^\ast$, we have:
:$\tuple {\map D {T^{\ast \ast \ast} }, T^{\ast \ast \ast} } = \tuple {\map D {\paren {\overline T}^\ast}, \paren {\overline T}^\ast}$
from Densely-Defined Linear Operator is Closable iff Adjoint is Densely-Defined.
== Sources ==
* : $\text {VIII}.1$: Domains, graphs, adjoints and spectrum
Category:Adjoints (Densely-Defined Linear Operators)
Category:Closable Densely-Defined Linear Operators
Adjoint of Composition of Linear Transformations is Composition of Adjoints

== Theorem ==
Let $\mathbb F \in \set {\R, \C}$.
Let $\HH$, $\KK$ and $\LL$ be a Hilbert spaces over $\mathbb F$.
Let $A : \KK \to \LL$ and $B : \HH \to \KK$ be bounded linear transformations.
Then:
:$\paren {A B}^* = B^* A^*$
where $^*$ denotes adjoining.
== Proof ==
Let ${\innerprod \cdot \cdot}_\HH$, ${\innerprod \cdot \cdot}_\KK$ and ${\innerprod \cdot \cdot}_\LL$ denote inner products over $\HH$, $\KK$ and $\LL$ respectively.
Let $h \in \HH$ and $l \in \LL$.
Then:
We also have, by the definition of the adjoint:
:$\innerprod {\map {\paren {A B} } h} l_\LL = \innerprod h {\map {\paren {A B}^*} l}_\LL$
So, by the uniqueness part of Existence and Uniqueness of Adjoint:
:$\paren {A B}^* = B^* A^*$
== Sources ==
* $\text {II}.2.6 \ \text {(b)}$
* $13.1$: Existence of Hilbert Adjoint
Category:Adjoints
Adjoint of Densely-Defined Linear Operator is Closed

== Theorem ==
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\tuple {\map D T, T}$ be a densely-defined linear operator.
Let $\tuple {\map D {T^\ast}, T^\ast}$ be the adjoint of $\tuple {\map D T, T}$.
Then $\tuple {\map D {T^\ast}, T^\ast}$ is closed.
== Proof ==
Let $\struct {\HH \times \HH, \innerprod \cdot \cdot_{\HH \times \HH} }$ be the Hilbert space direct sum of $\HH$ with itself.
From Equivalent Norms on Direct Product of Normed Vector Spaces, the inner product norm on $\struct {\HH \times \HH, \innerprod \cdot \cdot_{\HH \times \HH} }$ is equivalent to the direct product norm on $\HH \times \HH$.
Hence from Open Sets in Vector Spaces with Equivalent Norms Coincide it is enough to show that $\map \GG {T^\ast}$ is closed in $\struct {\HH \times \HH, \innerprod \cdot \cdot_{\HH \times \HH} }$.
Define $V : \HH \times \HH \to \HH \times \HH$ by:
:$\map V {x, y} = \tuple {-y, x}$
for each $\tuple {x, y} \in \HH \times \HH$.
From Graph of Adjoint of Densely-Defined Linear Operator, we have:
:$\map \GG {T^\ast} = \paren {V \sqbrk {\map \GG T} }^\bot$
where orthocomplementation is considered in the Hilbert space direct sum $\struct {\HH \times \HH, \innerprod \cdot \cdot_{\HH \times \HH} }$.
From Orthocomplement is Closed Linear Subspace, $\paren {V \sqbrk {\map \GG T} }^\bot$ is closed in $\struct {\HH \times \HH, \innerprod \cdot \cdot_{\HH \times \HH} }$.
Hence $\map \GG {T^\ast}$ is closed in $\struct {\HH \times \HH, \innerprod \cdot \cdot_{\HH \times \HH} }$.
So $\tuple {\map D {T^\ast}, T^\ast}$ is closed.
== Sources ==
* : $13.7$: Graphs
Category:Adjoints (Densely-Defined Linear Operators)
Adjoint of Densely-Defined Linear Operator reverses Extension of Mapping

== Theorem ==
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\tuple {\map D T, T}$ and $\tuple {\map D S, S}$ be densely-defined linear operators such that:
:$\tuple {\map D S, S}$ extends $\tuple {\map D T, T}$.
Let $\tuple {\map D {T^\ast}, T^\ast}$ and $\tuple {\map D {S^\ast}, S^\ast}$ be the adjoints of $\tuple {\map D T, T}$ and $\tuple {\map D S, S}$ respectively. 
Then $\tuple {\map D {T^\ast}, T^\ast}$ extends $\tuple {\map D {S^\ast}, S^\ast}$. 
== Proof ==
We first show that $\map D {S^\ast} \subseteq \map D {T^\ast}$.
For each $y \in \HH$, define $f_{y, T} : \map D T \to \C$ by:
:$\map {f_{y, T} } x = \innerprod {T x} y$ for each $x \in \map D T$.
Define $f_{y, S} : \map D S \to \C$ by:
:$\map {f_{y, S} } x = \innerprod {S x} y$ for each $x \in \map D S$.
Let $y \in \map D {S^\ast}$.
Then $f_{y, S}$ is a bounded linear functional.
Since $\tuple {\map D S, S}$ extends $\tuple {\map D T, T}$, we have $S x = T x$ for each $x \in \map D T$. 
Hence we have:
:$\map {f_{y, S} } x = \map {f_{y, T} } x$ for each $x \in \map D T$.
Hence $f_{y, T}$ is a bounded linear functional.
Hence $y \in \map D {T^\ast}$.
We have, by the construction of the adjoints, that $S^\ast y$ is the element of $\HH$ such that:
:$\innerprod {S x} y = \innerprod x {S^\ast y}$ for each $x \in \map D S$.
In particular for $x \in \map D T$ we have:
:$\innerprod {T x} y = \innerprod {S x} y = \innerprod x {S^\ast y}$
while we also have:
:$\innerprod {T x} y = \innerprod x {T^\ast y}$
Since $S^\ast y$ is unique, we have $S^\ast y = T^\ast y$.
Hence $\tuple {\map D {T^\ast}, T^\ast}$ extends $\tuple {\map D {S^\ast}, S^\ast}$.
Category:Adjoints (Densely-Defined Linear Operators)
Adjoint of Direct Sum of Bounded Linear Operators on Hilbert Space

== Theorem ==
Let $\GF \in \set {\R, \C}$. 
Let $\sequence {\family {\HH_i, \innerprod \cdot \cdot_i} }_{i \mathop \in I}$ be a $I$-indexed family of Hilbert spaces over $\GF$.
For each $i \in I$, let $T_i : \HH_i \to \HH_i$ be a bounded linear operator.
Suppose that:
:$\ds \sup_{i \mathop \in I} \norm {T_i}_{\map B {\HH_i} } < \infty$
where $\norm {\, \cdot \,}_{\map B {\HH_i} }$ is the norm of a bounded linear operator on $\HH_i$.
Let:
:$\ds \HH = \bigoplus_{i \mathop \in I} \HH_i$
be the Hilbert space direct sum of $\sequence {\family {\HH_i, \innerprod \cdot \cdot_i} }_{i \mathop \in I}$ with inner product $\innerprod \cdot \cdot$ and norm $\norm {\, \cdot \,}$.
Let:
:$\ds T = \bigoplus_{i \mathop \in I} T_i$
and:
:$\ds \bigoplus_{i \mathop \in I} T_i^\ast$ 
be the direct sum of $\family {T_i}_{i \mathop \in I}$ and $\family {T_i^\ast}_{i \mathop \in I}$ respectively.
Then:
:$\ds T^\ast = \bigoplus_{i \mathop \in I} T_i^\ast$
That is:
:$\ds \paren {\bigoplus_{i \mathop \in I} T_i}^\ast = \bigoplus_{i \mathop \in I} T_i^\ast$
== Proof ==
From Norm of Adjoint, we have:
:$\norm {T_i}_{\map B {\HH_i} } = \norm {T_i^\ast}_{\map B {\HH_i} }$
and hence:
:$\ds \sup_{i \mathop \in I} \norm {T_i^\ast}_{\map B {\HH_i} } < \infty$
so we can indeed define $\ds \bigoplus_{i \mathop \in I} T_i^\ast$.
Let $f, g \in \HH$. 
We then have:
and hence conclude:
:$\ds T^\ast = \bigoplus_{i \mathop \in I} T_i^\ast$
Category:Direct Sums of Hilbert Spaces
Adjoint of Direct Sum of Densely-Defined Linear Operators on Hilbert Spaces

== Theorem ==
Let $\family {\tuple {\HH_n, \innerprod \cdot \cdot_{\HH_n} } }_{n \mathop \in \N}$ be a sequence of Hilbert spaces over $\C$.
Let $\tuple {\HH, \innerprod \cdot \cdot_\HH}$ be the Hilbert space direct sum of $\family {\tuple {\HH_n, \innerprod \cdot \cdot_n} }_{n \mathop \in \N}$.
For each $n \in \N$, let $\tuple {\map D {T_n}, T_n}$ be a densely-defined linear operator with adjoint $\tuple {\map D {T_n^\ast}, T_n^\ast}$. 
Let $\ds \tuple {\map D T,T}$ be the direct sum of $\sequence {\tuple {\map D {T_n}, T_n} }_{n \mathop \in \N}$ with adjoint $\tuple {\map D {T^\ast}, T^\ast}$.
Let $\ds \tuple {\map D {\bigoplus_{n \mathop = 1}^\infty T_n^\ast}, \bigoplus_{n \mathop = 1}^\infty T_n^\ast}$ be the direct sum of $\sequence {\tuple {\map D {T_n^\ast}, T_n^\ast} }_{n \mathop \in \N}$.
Then we have:
:$\ds \tuple {\map D {T^\ast}, T^\ast} = \tuple {\map D {\bigoplus_{n \mathop = 1}^\infty T_n^\ast}, \bigoplus_{n \mathop = 1}^\infty T_n^\ast}$
== Proof ==
For each $\psi = \sequence {\psi_n}_{n \mathop \in \N} \in \HH$, define $f_\psi : \map D T \to \C$ by:
:$\map {f_\psi} \phi = \innerprod {T \phi} \psi_\HH$
for each $\phi \in \map D T$. 
Let $\ds \psi = \sequence {\psi_n}_{n \mathop \in \N} \in \map D {\bigoplus_{n \mathop = 1}^\infty T_n^\ast}$.
We have:
Hence $f_\psi$ is a bounded linear functional.
Hence $\psi \in \map D {T^\ast}$.
Hence:
:$\ds \map D {\bigoplus_{n \mathop = 1}^\infty T_n^\ast} \subseteq \map D {T^\ast}$ 
Further, by the uniqueness of the adjoint, we have that:
:$\ds T^\ast \psi = \paren {\bigoplus_{n \mathop = 1}^\infty T_n^\ast} \psi$
for each $\ds \psi \in \map D {\bigoplus_{n \mathop = 1}^\infty T_n^\ast}$.
To finish we need to show that:
:$\ds \map D {T^\ast} \subseteq \map D {\bigoplus_{n \mathop = 1}^\infty T_n^\ast}$
Let $\psi \in \map D {T^\ast}$.
From the definition of the adjoint, we want to show that $\psi_n \in \map D {T_n^\ast}$. 
We have that $f_\psi$ is a bounded linear functional.
Hence there exists $M > 0$ such that:
:$\cmod {\map {f_\psi} \phi} \le M \norm \phi_\HH$
for each $\phi \in \map D T$.
We first need to show that $\psi_n \in \map D {T_n^\ast}$ for each $n \in \N$.
Let $\phi \in \map D {T_n}$.
Define $\phi^{(n)} \in \HH$ by:
:$\map {\phi^{(n)} } n = \phi$
and:
:$\map {\phi^{(n)} } k = {\mathbf 0}_{\HH_k}$ for $k \ne n$.
Then $\phi^{(n)} \in \map D T$. 
We then have:
Hence $\psi_n \in \map D {T_n^\ast}$.
So $\ds \psi \in \map D {\bigoplus_{n \mathop = 1}^\infty T_n^\ast}$ as required.
Category:Direct Sum of Densely-Defined Linear Operators
Adjoint of Finite Rank Operator

== Theorem ==
Let $\GF \in \set {\R, \C}$. 
Let $\struct {\HH, \innerprod \cdot \cdot_\HH}$ and $\struct {\KK, \innerprod \cdot \cdot_\KK}$ be Hilbert spaces over $\GF$.
Let $T \in \map {B_{00} } {\HH, \KK}$ be a bounded finite rank operator.
Then:
:$T^* \in \map {B_{00} } {\KK, \HH}$
that is, the adjoint of $T$ is also a bounded finite rank operator.
== Proof ==
From Characterization of Finite Rank Operators, there exists $e_1, e_2, \ldots, e_n \in \HH$ and $g_1, g_2, \ldots, g_n \in \KK$ such that: 
:$\ds T x = \sum_{i \mathop = 1}^n \innerprod x {e_i}_\HH g_i$ 
for each $x \in \HH$. 
Then for each $x \in \HH$, $y \in \KK$ we have: 
From Existence and Uniqueness of Adjoint: Lemma 1, we therefore have: 
:$\ds T^\ast y = \sum_{i \mathop = 1}^n \innerprod y {g_i}_\KK e_i$
for each $y \in \KK$. 
So that: 
:$\ds T^\ast y \in \span \set {e_1, e_2, \ldots, e_n}$ 
for each $y \in \KK$. 
So we have that $T^\ast$ has finite rank.
== Sources ==
* $\text{II}.4$ Exercise $3$
Category:Finite Rank Operators
Category:Adjoints