Added a characterisation of L^p.

src/cat/cat/index.tex

@@ -5,3 +5,4 @@
 
 \input{./cat-func.tex}
 \input{./universal.tex}
+\input{./tensor.tex}

src/cat/cat/tensor.tex

@@ -0,0 +1,55 @@
+\section{The Tensor Product}
+\label{section:tensor-product}
+
+
+
+\begin{definition}[Tensor Product]
+\label{definition:tensor-product}
+    Let $R$ be a commutative ring and $\seqf{E_j}$ be $R$ modules, then there exists a pair $(\bigotimes_{j = 1}^n E_j, \iota)$ such that:
+    \begin{enumerate}
+        \item $\bigotimes_{j = 1}^n E_j$ is an $R$-module. 
+        \item $\iota: \prod_{j = 1}^n E_j \to \bigotimes_{j = 1}^n E_j$ is a $n$-linear map. 
+        \item[(U)] For any pair $(F, \lambda)$ satisfying (1) and (2), there exists a unique $\Lambda \in \hom(\bigotimes_{j = 1}^n E_j; F)$ such that the following diagram commutes:
+        \[
+        \xymatrix{
+        \prod_{j = 1}^n E_j \ar@{->}[rd]_{\lambda} \ar@{->}[r]^{\iota} & \bigotimes_{j = 1}^n E_j \ar@{->}[d]^{\Lambda} \\
+        & F
+        }
+        \] 
+
+        \item $\bigotimes_{j = 1}^n E_j$ is the linear span of $\iota(\prod_{j = 1}^n E_j)$. 
+    \end{enumerate}
+
+    The module $\bigotimes_{j = 1}^n E_j$ is the \textbf{tensor product} of $\seqf{E_j}$, and $\iota: \prod_{j = 1}^n E_j \to \bigotimes_{j = 1}^n E_j$ is the \textbf{canonical embedding}. For any $(x_1, \cdots, d_n) \in \prod_{j = 1}^n E_j$, the image
+    \[
+    x_1 \otimes \cdots \otimes x_n = \iota(x_1, \cdots, x_n)
+    \]
+
+    is its \textbf{tensor product}.
+     
+\end{definition}
+\begin{proof}
+    Let $M$ be the free module generated by $\prod_{j = 1}^nE_j$, and $N \subset M$ be the submodule generated by elements of the following form:
+    \begin{enumerate}
+        \item For any $1 \le j \le n$, $(x_1, \cdots, x_n) \in \prod_{k = 1}^n E_k$, and $x_j \in E_j$, 
+        \[
+        (x_1, \cdots, x_j + x_j', \cdots, x_n) - (x_1, \cdots, x_j, \cdots, x_n) - (x_1, \cdots, x_j', \cdots, x_n)
+        \]
+
+        \item For any $(x_1, \cdots, x_n) \in \prod_{k = 1}^n E_k$ and $\alpha \in R$, 
+        \[
+        (x_1, \cdots, \alpha x_j, \cdots, x_n) - \alpha(x_1, \cdots, x_n)
+        \]
+    \end{enumerate}
+    
+    (1), (2): Let $\bigotimes_{j = 1}^n E_j = M/N$ and
+    \[
+    \iota: \prod_{j = 1}^n E_j \to \bigotimes_{j = 1}^n E_j \quad (x_1, \cdots, x_n) \mapsto (x_1, \cdots, x_n) + N
+    \]
+
+    then by definition of $N$, $\iota$ is $n$-linear. 
+
+    (U): Let $(F, \lambda)$ be a pair satisfying (1) and (2), then $\lambda$ admits a unique extension to a linear map $\ol \lambda: M \to F$. Since $\lambda$ is $n$-linear, $\ker \ol \lambda \supset N$. By the first isomorphism theorem, there exists a unique $\Lambda \in \hom(\bigotimes_{j = 1}^n E_j; F)$ such that the given diagram commutes. 
+
+    (4): Since $M$ is the free module generated by $\prod_{j = 1}^n E_j$, $M/N$ is generated by $\iota(\prod_{j = 1}^n E_j)$.  
+\end{proof}

src/cat/cat/universal.tex

@@ -209,7 +209,7 @@ Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim
 
 \begin{proposition}
 \label{proposition:module-inverse-limit}
-    Let $R$ be a ring and $(\seqi{A}, \bracs{T^i_j|i, j \in I, i \lesssim j)}$ be a downward-directed system of $R$-modules, then there exists $(A, \bracsn{T^A_i}_{i \in I})$ such that:
+    Let $R$ be a ring and $(\seqi{A}, \bracs{T^i_j|i, j \in I, i \lesssim j})$ be a downward-directed system of $R$-modules, then there exists $(A, \bracsn{T^A_i}_{i \in I})$ such that:
     \begin{enumerate}
         \item For each $i \in I$, $T^A_i \in \hom(A; A_i)$.
         \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
@@ -253,3 +253,6 @@ Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim
 
     so $S \in \hom(B; A)$, and the diagram commutes. Since any map $f: B \to A$ is uniquely determined by its composition with the projections, $S$ is unique.
 \end{proof}
+
+
+

src/fa/lc/convex.tex

@@ -7,6 +7,28 @@
     Let $E$ be a vector space over $K \in \RC$, then $A \subset E$ is \textbf{convex} if for any $x, y \in A$, $\bracs{\lambda x + (1 - \lambda) y| \lambda \in [0, 1]} \subset A$.
 \end{definition}
 
+\begin{definition}[Convex Hull]
+\label{definition:convex-hull}
+    Let $E$ be a vector space over $K \in \RC$ and $A \subset E$, then the set
+    \[
+    \text{Conv}(A) = \bracs{\sum_{j = 1}^n t_j x_j \bigg | \seqf{t_j} \subset [0, 1], \seqf{x_j} \subset E, \sum_{j = 1}^n t_j = 1 }
+    \]
+
+    is the \textbf{convex hull} of $A$. 
+\end{definition}
+
+\begin{definition}[Convex Circled Hull]
+\label{definition:convex-circled-hull}
+    Let $E$ be a vector space over $K \in \RC$ and $A \subset E$, then the set
+    \[
+    \Gamma(A) = \bracs{\sum_{j = 1}^n t_j x_j \bigg | \seqf{t_j} \subset K, \seqf{x_j} \subset E, \sum_{j = 1}^n |t_j| \le 1 }
+    \]
+
+    is the \textbf{convex circled hull} of $A$. 
+\end{definition}
+
+
+
 \begin{lemma}[{{\cite[II.1.1]{SchaeferWolff}}}]
 \label{lemma:convex-interior}
     Let $E$ be a TVS over $K \in \RC$, $A \subset E$ be convex, $x \in A^o$, and $y \in \ol{A}$, then
@@ -67,10 +89,6 @@
 
 
 
-
-
-
-
 \begin{definition}[Sublinear Functional]
 \label{definition:sublinear-functional}
     Let $E$ be a vector space over $K \in \RC$, then a \textbf{sublinear functional} is a mapping $\rho: E \to \real$ such that:

src/fa/lc/index.tex

@@ -10,3 +10,4 @@
 \input{./inductive.tex}
 \input{./hahn-banach.tex}
 \input{./spaces-of-linear.tex}
+\input{./tensor.tex}

src/fa/lc/tensor.tex

@@ -0,0 +1,171 @@
+\section{The Projective Tensor Product}
+\label{section:projective-tensor-product}
+
+\begin{definition}[Projective Tensor Product]
+\label{definition:projective-tensor-product}
+    Let $E, F$ be locally convex spaces over $K \in \RC$, then there exists a pair $(E \otimes_\pi F, \iota)$ such that:
+    \begin{enumerate}
+        \item $E \otimes_\pi F$ is a locally convex space over $K$. 
+        \item $\iota \in L^2(E, F; E \otimes_\pi F)$ is a continuous bilinear map. 
+        \item[(U1)] For any $(G, \lambda)$ satisfying (1) and (2), there exists a unique $\Lambda \in L(E \otimes_\pi F; G)$ such that the following diagram commutes:
+        \[
+        \xymatrix{
+        E \times F \ar@{->}[rd]_{\lambda} \ar@{->}[r]^{\iota} & E \otimes F \ar@{->}[d]^{\Lambda} \\
+        & G
+        }
+        \]
+        \item[(U2)] For any topology $\topo$ on $E \otimes_\pi F$ satisfying (1) and (2), $\topo$ is coarser than the topology on $E \otimes_\pi F$. 
+
+        \item $E \otimes_\pi F$ is the linear span of $\iota(E \times F)$. 
+        \item For any $U \subset E$ and $V \subset F$, let $U \otimes V = \bracs{u \otimes v|u \in U, v \in V}$, then the convex circled hulls
+        \[
+        \fB = \bracsn{\Gamma(U \otimes V)| U \in \cn_E(0), V \in \cn_F(0)}
+        \]
+
+        is a fundamental system of neighbourhoods at $0$ for $E \otimes_\pi F$.
+    \end{enumerate}
+    
+    The space $E \otimes_\pi F$ is the \textbf{projective tensor product} of $E$ and $F$, and the mapping $\iota \in L^2(E, F; E \otimes_\pi F)$ is the \textbf{canonical embedding}.
+
+    The space $E \widetilde{\otimes}_\pi F$ denotes the Hausdorff completion of $E \otimes_\pi F$.
+\end{definition}
+\begin{proof}
+    Let $E \otimes_\pi F = E \otimes F$ be the \hyperref[tensor product]{definition:tensor-product} of $E$ and $F$ as vector spaces. Let $\mathscr{T} \subset 2^{2^X}$ be the collection of all locally convex topologies satisfying (1) and (2), and let $\mathcal{S}$ be the projective topology on $E \otimes_\pi F$ generated by $\mathscr{T}$. 
+    
+    (1): By \autoref{proposition:lc-projective-topology}, $\mathcal{S}$ is a locally convex topology on $E \otimes_\tau F$. 
+
+    (2): Since $\iota: E \times F \to E \otimes_\pi F$ is continuous with respect to every topology in $\mathscr{T}$, it is also continuous with respect to $\mathcal{S}$. 
+
+    (U2): Since $\mathcal{T} \in \mathscr{T}$, $\mathcal{S} \supset \mathcal{T}$.
+
+    (U1): By (U) of the \hyperref[tensor product]{definition:tensor-product}, there exists a unique $\Lambda \in \hom(E \otimes_\pi F; G)$ such that the given diagram commutes. Since $\lambda$ is continuous, the projective topology generated by $\Lambda$ satisfies (1) and (2). By (U2), $\mathcal{S}$ contains the projective topology generated by $\Lambda$. Therefore $\Lambda \in L(E \otimes_\pi; F)$. 
+
+    (5): By (4) of the \hyperref[tensor product]{definition:tensor-product}. 
+
+    (6): Let $U \in \cn_E(0)$ and $V \in \cn_F(0)$ be balanced. For any $\sum_{j = 1}^n x_j \otimes y_j \in E \otimes_\pi F$, then there exists $\lambda > 0$ such that $\seqf{x_j} \subset \lambda U$ and $\seqf{y_j} \subset \lambda V$. In which case, $\sum_{j = 1}^n x_j \otimes y_j \subset \lambda \Gamma (U \otimes V)$, so $\fB$ is a collection of convex, circled, and radial sets. Since $\fB$ defines a locally convex topology that satisfies (1) and (2), $\mathcal{S}$ contains the topology defined by $\fB$. 
+
+    On the other hand, for any convex and circled set $W \in \cn_{E \otimes_\pi F}(0)$, there exists $U \in \cn_E(0)$ and $V \in \cn_F(0)$ such that $U \otimes V \subset W$. In which case, $W \supset \Gamma(U \otimes V)$, so $\fB$ is a fundamental system of neighbourhoods at $0$ for $E \otimes_\pi F$. 
+\end{proof}
+
+\begin{definition}[Cross Seminorm, {{\cite[III.6.3]{SchaeferWolff}}}]
+\label{definition:cross-seminorm}
+    Let $E, F$ be locally convex spaces over $K \in \RC$. For any convex circled sets $U \in \cn_E(0)$ and $V \in \cn_F(0)$, let $p: E \to [0, \infty)$ and $q: F \to [0, \infty)$ be their \hyperref[gauges]{definition:gauge}. For any $z \in E \otimes_\pi F$, let
+    \[
+    \rho(z) = \inf\bracs{\sum_{j = 1}^n p(x_j)q(y_j) \bigg | \seqf{(x_j,y_j)} \subset E \times F, z = \sum_{j = 1}^n x_j \otimes y_j}
+    \]
+
+    then
+    \begin{enumerate}
+        \item $\rho$ is a continuous seminorm on $E \otimes_\pi F$.
+        \item $\rho$ is the gauge of $\Gamma(U \otimes V)$. 
+        \item For any $x \in E$ and $y \in F$, $\rho(x \otimes y) = p(x)q(Y)$. 
+        \item $\rho$ is a norm if and only if $[\cdot]_U$ and $[\cdot]_V$ are norms. 
+    \end{enumerate}
+
+    and the seminorm $\rho = p \otimes q$ is the \textbf{cross seminorm} of $p$ and $q$. Moreover, 
+    \begin{enumerate}
+        \item[(5)] If the seminorms $\seqi{p}$ define the topology on $E$, and the seminorms $\seqj{q}$ define the topology on $F$, then the seminorms $\bracsn{p_i \otimes q_j| (i, j) \in I \times J}$ define the topology on $E \otimes_\pi F$. 
+    \end{enumerate}
+    
+\end{definition}
+\begin{proof}
+    (1): Let $\lambda \in K$, then for any $\seqf{(x_j,y_j)} \subset E \times F$, 
+    \[
+    |\lambda| \sum_{j = 1}^n p(x_j)q(y_j) = \sum_{j = 1}^n p(\lambda x_j)q(y_j)
+    \]
+
+    and
+    \[
+    \lambda\sum_{j = 1}^n x_j \otimes y_j =  \sum_{j = 1}^n \lambda x_j \otimes y_j
+    \]
+
+    so for any $z \in E \otimes_\pi F$, $|\lambda|\rho(z) = \rho(\lambda z)$. 
+    
+    Let $z, z' \in E \otimes F$, $\seqf{(x_j,y_j)}, \bracsn{(x_j',y_j')}_1^m \subset E \times F$ such that $z = \sum_{j = 1}^n x_j \otimes y_j$ and $z' = \sum_{j = 1}^m x_j' \otimes y_j'$, then
+    \[
+    z + z' = \sum_{j = 1}^n x_j \otimes y_j + \sum_{j = 1}^m x_j' \otimes y_j'
+    \]
+
+    and
+    \[
+    \rho(z + z') \le \sum_{j = 1}^n p(x_j)q(y_j) + \sum_{j = 1}^m p(x_j')q(y_j')
+    \]
+
+    so $\rho$ satisfies the triangle inequality. 
+
+    (2): Let $z \in \Gamma(U \otimes V)$, then there exists $\seqf{(x_j, y_j)} \subset U \times V$ and $\seqf{\lambda_j} \subset K$ such that $\sum_{j = 1}^n |\lambda_j| \le 1$ and $z = \sum_{j = 1}^n \lambda x_j \otimes y_j$. In which case, 
+    \begin{align*}
+        \rho(z) &\le \sum_{j = 1}^n p(\lambda x_j)q(y_j) = \sum_{j = 1}^n |\lambda_j|p(x_j)q(y_j) \\
+        &< \sum_{j = 1}^n |\lambda_j| \le 1
+    \end{align*}
+
+    so $\Gamma(U \otimes V) \subset \bracs{\rho < 1}$.
+
+    Let $z \in \bracs{\rho < 1}$, then there exists $\seqf{(x_j, y_j)} \subset E \times F$ such that $z = \sum_{j = 1}^nx_j \otimes y_j$ and $\sum_{j = 1}^n p(x_j)q(x_j) < 1$. Let $\eps > 0$ such that $\sum_{j = 1}^n(p(x_j) + \eps)(q(x_j) + \eps) < 1$, then
+    \[
+    z = \sum_{j = 1}^n (p(x_j) + \eps)(q(x_j) + \eps) \cdot \underbrace{\frac{x_j}{p(x_j) + \eps}}_{\in \bracs{p < 1} = U} \otimes \underbrace{\frac{y_j}{q(x_j) + \eps}}_{\in \bracs{q < 1} = V} \in \Gamma(U \otimes V)
+    \]
+
+    and $\Gamma(U \otimes V) \supset \bracs{\rho < 1}$.
+    
+    (3): Let $x \in U$ and $y \in V$. By the \hyperref[Hahn-Banach Theorem]{proposition:hahn-banach-utility}, there exists $\phi \in E^*$ and $\psi \in F^*$ such that $\dpn{x, \phi}{E} = p(x)$, $\dpn{y, \psi}{F} = q(x)$, $|\phi| \le p$, and $|\psi| \le q$. By (U1) of the \hyperref[projective tensor product]{definition:projective-tensor-product}, there exists $\Phi \in (E \otimes_\pi F)^*$ such that the following diagram commutes
+    \[
+    \xymatrix{
+    E \times F \ar@{->}[rd]_{\phi \cdot \psi} \ar@{->}[r]^{\iota} & E \otimes_\pi F \ar@{->}[d]^{\Phi} \\
+    & K
+    }
+    \]
+    
+    For any $z \in E \otimes_\pi F$ and $\seqf{(x_j, y_j)} \subset E \times F$ such that $z = \sum_{j = 1}^n x_j \otimes y_j$, 
+    \[
+    \Phi(z) = \sum_{j = 1}^n \Phi(x_j \otimes y_j) = \sum_{j = 1}^n \phi(x_j)\psi(y_j) \le \sum_{j = 1}^n p(x_j)q(y_j)
+    \]
+
+    As the above holds for all such $\seqf{(x_j, y_j)} \subset E \times F$, $|\Phi| \le \rho$. Since $\Phi(x \otimes y) = p(x)q(y)$, $\rho(x \otimes y) = p(x)q(y)$ as well. 
+
+    (5): By (6) of \autoref{definition:projective-tensor-product}. 
+\end{proof}
+
+\begin{theorem}[{{\cite[III.6.4]{SchaeferWolff}}}]
+\label{theorem:metrisable-tensor-product}
+    Let $E, F$ be metrisable locally convex spaces over $K \in \RC$, then for any $z \in E \td{\otimes}_\pi F$, there exists $\seq{\lambda_n} \subset K$ and $\seq{(x_j, y_j)} \subset E \times F$ such that:
+    \begin{enumerate}
+        \item $\sum_{n \in \natp}|\lambda_n| < \infty$. 
+        \item $\limv{n}x_n = 0$ and $\limv{n}y_n = 0$.
+        \item $z = \sum_{n = 1}^\infty \lambda_n x_n \otimes y_n$. 
+    \end{enumerate}
+    
+\end{theorem}
+\begin{proof}
+    Let $\seq{p_n}$ and $\seq{q_n}$ be increasing sequences of continuous seminorms that induce the topology on $E$ and $F$, respectively. For each $n \in \natp$, let $r_n = p_n \otimes q_n$, and $\td r_n$ be the continuous extension of $r_n$ to $E \td{\otimes}_\pi F$. 
+
+    Let $u \in E \td{\otimes}_\pi F$, then there exists $\seq{u_n} \subset E \otimes_\pi F$ such that $\td r_n(u - u_n) < 2^{-n}/n^2$ for all $n \in \natp$. For each $N \in \natp$, let $v_N = u_{N+1} - u_N$, then
+    \begin{align*}
+        r_N(v_N) &= \td r_n(u_{N+1} - u_n) \le \td r_N(u - u_N) + \td r_{N}(u - u_{N+1}) \\
+        &\le \td r_N(u - u_N) + \td r_{N+1}(u - u_{N+1}) < 2^{-N+1}/n^2
+    \end{align*}
+
+    Since $r_N = p_N \otimes q_N$, there exists $\bracsn{(x_{N, k}, y_{N, k})}_{1}^{n_N} \subset X \times Y$ such that $v_N = \sum_{k = 1}^{n_N}x_{N, k} \otimes y_{N, k}$ and
+    \[
+        r_N(v_N) = \sum_{k = 1}^{n_N}p_N(x_{N, k})q_N(x_{N, k}) < 2^{-N+1}/n^2
+    \]
+    
+    By rescaling, assume without loss of generality that there exists $\bracsn{\lambda_{N, k}}_1^{n_N}$ such that\
+    \begin{enumerate}
+        \item $v_N = \sum_{k = 1}^{n_N}\lambda_{N, k}x_{N, k} \otimes y_{N, k}$.
+        \item For each $1 \le k \le n_N$, $p_N(x_{N, k}), q_N(x_{N, k}) \le 1/M$.
+        \item $\sum_{k = 1}^{n_N}|\lambda_k| \le 2^{-N+2}$.
+    \end{enumerate}
+    
+    From here, let $\seqf{(x_j, y_j)} \subset X \times Y$ such that $u_1 = \sum_{j = 1}^n x_j \otimes y_j$, then
+    \[
+    u = u_1 + \sum_{N = 1}^\infty v_N = \sum_{j = 1}^n x_j \otimes y_j + \sum_{N = 1}^\infty \sum_{k = 1}^{n_N}\lambda_{N, k}x_{N, k} \otimes y_{N, k}
+    \]
+
+    where $x_{N, k} \to 0$ and $y_{N, k} \to 0$ as $N \to \infty$, and $\sum_{N \in \natp}\sum_{k = 1}^{n_N}|\lambda_{N, k}| < \infty$.
+    
+\end{proof}
+
+
+
+

src/fa/lp/definition.tex

@@ -127,6 +127,46 @@
     Let $f \in L^p(X; E)$. By \autoref{definition:strongly-measurable}, there exists $\seq{f_n} \subset \Sigma(X, \cm; E)$ such that $\norm{f_n}_E \le \norm{f}_E$ and $\norm{f_n - f}_E \to 0$ strongly pointwise as $n \to \infty$. By the \hyperref[Dominated Convergence Theorem]{proposition:dct-lp}, $f_n \to f$ in $L^p(X; E)$.
 \end{proof}
 
+\begin{theorem}[{{\cite[III.6.5]{SchaeferWolff}}}]
+\label{theorem:l1-tensor}
+    Let $(X, \cm, \mu)$ be a measure space and $E$ be a Banach space over $\real \in \RC$, then the map $L^1(X; \real) \td{\otimes}_\mu E \to L^1(X; E)$ defined by extending
+    \[
+    L^1(X; \real) \times E \to L^1(X; E) \quad f \otimes x \mapsto x \cdot f
+    \]
+    
+    is an isometric isomorphism. 
+\end{theorem}
+\begin{proof}
+    By (U) of the \hyperref[tensor product]{definition:tensor-product}, the given map admits a unique extension
+    \[
+    M: L^1(X; \real) \otimes E \to L^1(X; E) \quad \sum_{j = 1}^n f_j \otimes x_j \mapsto \sum_{j = 1}^n x_j \cdot f_j
+    \]
+
+    Restricting $M$ to the simple functions yields a linear isomorphism
+    \[
+    M: [L^1(X; \real) \cap \Sigma(X; \real)] \otimes E \to L^1(X; E) \cap \Sigma(X; E)
+    \]
+
+    For any $\phi \in L^1(X; E) \cap \Sigma(X; E)$, write
+    \[
+    \phi = \sum_{y \in \phi(X) \setminus \bracs{0}}y \cdot \one_{\bracs{\phi = y}} = M\braks{\sum_{y \in \phi(X) \setminus \bracs{0}}\one_{\bracs{\phi = y}} \otimes y}
+    \]
+
+    then
+    \[
+    \normn{M^{-1}\phi}_{L^1(X; \real) \otimes E} \le \sum_{y \in \phi(X) \setminus \bracs{0}} \norm{y}_E \cdot \mu\bracs{\phi = y} = \int \norm{\phi}_E d\mu = \norm{\phi}_{L^1(X; E)}
+    \]
+
+    On the other hand, for any representation $M^{-1}\phi = \sum_{j = 1}^n a_j \one_{A_j}$, 
+    \[
+    \normn{\phi}_{L^1(X; E)} \le \sum_{j = 1}^n \norm{a_j}_E \mu(A_j) = \sum_{j = 1}^n \norm{a_j}_E \normn{\one_{A_j}}_{L^1(X; \real)}
+    \]
+
+    As this holds for all such representations, $\normn{\phi}_{L^1(X; E)} = \normn{M^{-1}\phi}_{L^1(X; \real) \otimes E}$. Therefore $M$ restricted to $[L^1(X; \real) \cap \Sigma(X; \real)] \otimes E$ is an isometry. By \autoref{proposition:lp-simple-dense}, $[L^1(X; \real) \cap \Sigma(X; \real)] \otimes E$ is dense in $L^1(X; \real) \widehat{\otimes}_\pi E$, and $L^1(X; E) \cap \Sigma(X; E)$ is dense in $E$. By the \hyperref[Linear Extension Theorem]{theorem:linear-extension-theorem-normed}, $M$ extends uniquely into the given map on $L^1(X; \real) \otimes E$, which then extends into an isometry $L^1(X; \real) \otimes E \to L^1(X; E)$. 
+\end{proof}
+
+
+
 \begin{theorem}[Markov's Inequality]
 \label{theorem:markov-inequality}
     Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, and $f: X \to E$ be a Borel measurable function, then

src/measure/bochner-integral/bochner.tex

@@ -44,6 +44,8 @@
     
 \end{proof}
 
+
+
 \begin{theorem}[Dominated Convergence Theorem]
 \label{theorem:dct-bochner}
     Let $(X, \cm, \mu)$ be a measure spacs, $E$ be a Banach space over $K \in \RC$, $\seq{f_n} \subset L^1(X; E)$, and $f \in L^1(X; E)$. If
@@ -58,3 +60,48 @@
     By the classical \hyperref[Dominated Convergence Theorem]{proposition:dct-lp}, $f_n \to f$ in $L^1(X; E)$. Since $h \mapsto \int h d\mu$ is a bounded linear operator, $\int f d\mu = \limv{n}\int f_n d\mu$. 
 \end{proof}
 
+\subsection{Vector Measure Version}
+\label{subsection:bochner-vector}
+
+
+
+\begin{definition}[Bochner Integral]
+\label{definition:bochner-integral-vector}
+    Let $(X, \cm)$ be a measurable space, $E, F$ be normed spaces over $K \in \RC$, $G$ be a Banach space over $K$,
+    \[
+    \lambda: E \times F \to G \quad (x, y) \mapsto xy
+    \]
+
+    be a bounded bilinear map, and $\mu: \cm \to F$ be a vector measure, then there exists a unique $I_\lambda \in L(L^1(X, |\mu|; E); G)$ such that:
+    \begin{enumerate}
+        \item For any $x \in E$ and $A \in \cm$, $I_\lambda(x \cdot \one_A) = x \mu(A)$.
+        \item For any $f \in L^1(X, |\mu|; E)$, $\normn{I_\lambda f}_{G} \le \norm{\lambda}_{L^2(E, F; G)} \cdot \norm{f}_{L^1(X, |\mu|; E)}$.
+    \end{enumerate}
+    
+    For any $f \in L^1(X; E)$, $I_\lambda f = \int f d\lambda\mu$ is the \textbf{Bochner integral} of $f$ with respect to $\mu$ and $\lambda$. 
+\end{definition}
+\begin{proof}
+    Same as \autoref{definition:bochner-integral}.
+\end{proof}
+
+
+\begin{theorem}[Dominated Convergence Theorem]
+\label{theorem:dct-bochner-vector}
+    Let $(X, \cm, \mu)$ be a measure space, $E, F$ be normed spaces over $K \in \RC$, $G$ be a Banach space over $K$, 
+    \[
+    \lambda: E \times F \to G \quad (x, y) \mapsto xy
+    \]
+
+    be a bounded bilinear map, $\mu: \cm \to F$ be a vector measure, $\seq{f_n} \subset L^1(X, |\mu|; E)$, and $f \in L^1(X, |\mu|; E)$. If
+    \begin{enumerate}
+        \item[(a)] $f_n \to f$ strongly pointwise.
+        \item[(b)] There exists $g \in L^1(X) \cap L^+(X)$ such that $\norm{f_n}_E \le g$ for all $n \in \natp$. 
+    \end{enumerate}
+    
+    then $\int f d\lambda\mu = \limv{n}\int f_n d\lambda\mu$. 
+\end{theorem}
+\begin{proof}
+    By the classical \hyperref[Dominated Convergence Theorem]{proposition:dct-lp}, $f_n \to f$ in $L^1(X, |\mu|; E)$. Since $h \mapsto \int h d\lambda\mu$ is a bounded linear operator, $\int f d\lambda\mu = \limv{n}\int f_n d\lambda\mu$. 
+\end{proof}
+
+