diff --git a/preamble.sty b/preamble.sty index 731529c..15532fe 100644 --- a/preamble.sty +++ b/preamble.sty @@ -224,3 +224,8 @@ % Real or Complex Numbers \newcommand{\RC}{\bracs{\real, \complex}} + +% Convex Stuff +\newcommand{\conv}{\text{Conv}} +\newcommand{\aconv}{\text{AbsConv}} + diff --git a/src/fa/lc/convex.tex b/src/fa/lc/convex.tex index a17c58d..08b30ca 100644 --- a/src/fa/lc/convex.tex +++ b/src/fa/lc/convex.tex @@ -21,7 +21,7 @@ \label{definition:convex-circled-hull} Let $E$ be a vector space over $K \in \RC$ and $A \subset E$, then the set \[ - \Gamma(A) = \bracs{\sum_{j = 1}^n t_j x_j \bigg | \seqf{t_j} \subset K, \seqf{x_j} \subset E, \sum_{j = 1}^n |t_j| \le 1 } + \aconv(A) = \bracs{\sum_{j = 1}^n t_j x_j \bigg | \seqf{t_j} \subset K, \seqf{x_j} \subset E, \sum_{j = 1}^n |t_j| \le 1 } \] is the \textbf{convex circled hull} of $A$. diff --git a/src/fa/lc/spaces-of-linear.tex b/src/fa/lc/spaces-of-linear.tex index f64b663..67b6c3d 100644 --- a/src/fa/lc/spaces-of-linear.tex +++ b/src/fa/lc/spaces-of-linear.tex @@ -3,20 +3,15 @@ \begin{proposition} \label{proposition:lc-spaces-linear-map} - Let $T$ be a set, $\sigma \subset 2^T$ be an ideal, and $E$ be a locally convex space over $K$. For each $S \in \sigma$ and continuous seminorm $\rho: E \to [0, \infty)$, let - \[ - \rho_S: E^T \to [0, \infty] \quad f \mapsto \sup_{x \in S}\rho(f(x)) - \] - - then the the $\sigma$-uniform topology on $E^T$ is defined by distances of the form - \[ - d_{S, \rho}: E^T \times E^T \to [0, \infty] \quad (f, g) \mapsto \rho_S(f - g) - \] - - In particular, if $\cf \subset E^T$ is a subspace such that + Let $T$ be a set, $\sigma \subset 2^T$ be an ideal, $E$ be a locally convex space over $K$, and $\cf \subset E^T$ be a subspace such that \begin{enumerate} \item[(B)] For each $f \in \cf$ and $S \in \sigma$, $f(S) \subset E$ is bounded. \end{enumerate} + + For each $S \in \sigma$ and continuous seminorm $\rho: E \to [0, \infty)$, let + \[ + \rho_S: E^T \to [0, \infty] \quad f \mapsto \sup_{x \in S}\rho(f(x)) + \] then the $\sigma$-uniform topology on $\cf$ is induced by seminorms of the form $\rho_S$, where $\rho$ is a continuous seminorm on $E$, and $S \in \sigma$. In which case, the $\sigma$-uniform topology on $\cf$ is locally convex. \end{proposition} diff --git a/src/fa/lc/tensor.tex b/src/fa/lc/tensor.tex index 2e470dc..f5dd8e3 100644 --- a/src/fa/lc/tensor.tex +++ b/src/fa/lc/tensor.tex @@ -19,7 +19,7 @@ \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 \[ - \fB = \bracsn{\Gamma(U \otimes V)| U \in \cn_E(0), V \in \cn_F(0)} + \fB = \bracsn{\aconv(U \otimes V)| U \in \cn_E(0), V \in \cn_F(0)} \] is a fundamental system of neighbourhoods at $0$ for $E \otimes_\pi F$. @@ -42,9 +42,9 @@ (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 $\fB$. + (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, $\sum_{j = 1}^n x_j \otimes y_j \subset \lambda \aconv (U \otimes V)$, 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 $E \otimes_\pi F$. + 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, $W \supset \aconv(U \otimes V)$, so $\fB$ is a fundamental system of neighbourhoods at $0$ for $E \otimes_\pi F$. \end{proof} \begin{remark} @@ -64,7 +64,7 @@ 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 $\rho$ is the gauge of $\aconv(U \otimes V)$. \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} @@ -100,20 +100,20 @@ 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, + (2): Let $z \in \aconv(U \otimes V)$, 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 $\Gamma(U \otimes V) \subset \bracs{\rho < 1}$. + so $\aconv(U \otimes V) \subset \bracs{\rho < 1}$. 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 \[ - z = \sum_{j = 1}^n (p(x_j) + \eps)(q(x_j) + \eps) \cdot \underbrace{\frac{x_j}{p(x_j) + \eps}}_{\in \bracs{p < 1} = U} \otimes \underbrace{\frac{y_j}{q(x_j) + \eps}}_{\in \bracs{q < 1} = V} \in \Gamma(U \otimes V) + z = \sum_{j = 1}^n (p(x_j) + \eps)(q(x_j) + \eps) \cdot \underbrace{\frac{x_j}{p(x_j) + \eps}}_{\in \bracs{p < 1} = U} \otimes \underbrace{\frac{y_j}{q(x_j) + \eps}}_{\in \bracs{q < 1} = V} \in \aconv(U \otimes V) \] - and $\Gamma(U \otimes V) \supset \bracs{\rho < 1}$. + and $\aconv(U \otimes V) \supset \bracs{\rho < 1}$. (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 \[ diff --git a/src/fa/notation.tex b/src/fa/notation.tex index 19ada12..b774cad 100644 --- a/src/fa/notation.tex +++ b/src/fa/notation.tex @@ -20,7 +20,7 @@ $\widehat{E}$ & Hausdorff completion of TVS $E$. & \autoref{definition:tvs-completion} \\ % ---- Locally Convex ---- $\mathrm{Conv}(A)$ & Convex hull of $A$. & \autoref{definition:convex-hull} \\ - $\Gamma(A)$ & Convex circled hull of $A$. & \autoref{definition:convex-circled-hull} \\ + $\aconv(A)$ & Convex circled hull of $A$. & \autoref{definition:convex-circled-hull} \\ $[\cdot]_A$ & Gauge of a radial set $A$. & \autoref{definition:gauge} \\ $\rho_M$ & Quotient of seminorm $\rho$ by subspace $M$. & \autoref{definition:quotient-norm} \\ $E \otimes_\pi F$ & Projective tensor product of $E$ and $F$. & \autoref{definition:projective-tensor-product} \\