diff --git a/src/fa/lc/convex.tex b/src/fa/lc/convex.tex index 369f131..25cc1b3 100644 --- a/src/fa/lc/convex.tex +++ b/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: diff --git a/src/fa/lc/index.tex b/src/fa/lc/index.tex index 5628e2c..77ece48 100644 --- a/src/fa/lc/index.tex +++ b/src/fa/lc/index.tex @@ -10,3 +10,4 @@ \input{./inductive.tex} \input{./hahn-banach.tex} \input{./spaces-of-linear.tex} +\input{./tensor.tex} diff --git a/src/fa/lc/tensor.tex b/src/fa/lc/tensor.tex new file mode 100644 index 0000000..af4fa69 --- /dev/null +++ b/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} + + + + diff --git a/src/fa/lp/definition.tex b/src/fa/lp/definition.tex index 546ac47..828f67e 100644 --- a/src/fa/lp/definition.tex +++ b/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