From 968fbe6eba5e3103bd2c5ee3ecb8e905951b8cbc Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Sat, 27 Jun 2026 13:17:45 -0400 Subject: [PATCH] Added the support function. --- src/fa/convex/index.tex | 3 +- src/fa/convex/support.tex | 135 ++++++++++++++++++++++++++++++++++ src/fa/duality/polar.tex | 9 --- src/fa/notation.tex | 2 + src/fa/tvs/equicontinuous.tex | 4 +- 5 files changed, 141 insertions(+), 12 deletions(-) create mode 100644 src/fa/convex/support.tex diff --git a/src/fa/convex/index.tex b/src/fa/convex/index.tex index 60a43c3..8347bd8 100644 --- a/src/fa/convex/index.tex +++ b/src/fa/convex/index.tex @@ -3,4 +3,5 @@ \input{./def.tex} \input{./del.tex} -\input{./legendre.tex} \ No newline at end of file +\input{./legendre.tex} +\input{./support.tex} \ No newline at end of file diff --git a/src/fa/convex/support.tex b/src/fa/convex/support.tex new file mode 100644 index 0000000..cf4300c --- /dev/null +++ b/src/fa/convex/support.tex @@ -0,0 +1,135 @@ +\section{Support Functions} +\label{section:support-function} + +\begin{definition}[Support Function] +\label{definition:support-function} + Let $\dpn{E, F}{\lambda}$ be a duality over $\real$ and $A \subset E$ be non-empty, then the mapping + \[ + H_A: F \to (-\infty, \infty] \quad y \mapsto \sup_{x \in A}\dpn{x, y}{\lambda} + \] + + the \textbf{support function} of $A$ with respect to $\dpn{E, F}{\lambda}$. +\end{definition} + +\begin{definition}[Indicator Function] +\label{definition:infinity-characteristic-function} + Let $E$ be a vector space over $\real$ and $A \subset E$, then the mapping + \[ + I_A: E \to (-\infty, \infty] \quad x \mapsto \begin{cases} + \infty &x \not\in A \\ + 0 & x \in A + \end{cases} + \] + + is the \textbf{infinity characteristic function/indicator function} of $A$. +\end{definition} + + + +\begin{lemma} +\label{lemma:support-function-gymnastics} + Let $\dpn{E, F}{\lambda}$ be a duality over $\real$, then: + \begin{enumerate} + \item For any non-empty $A \subset E$, let $\ol{\conv}(A)$ be the $\sigma(E, F)$-closed convex hull of $A$, then $H_A = H_{\ol{\conv}(A)}$. + \item For any non-empty $A, B \subset E$, $H_A \le H_B$ if and only if $A$ is contained in the $\sigma(E, F)$-closed convex hull of $B$. + \item For any non-empty $A \subset E$, $I_A^* = H_A$. + \item For any $A \subset E$ non-empty, $\sigma(E, F)$-closed, and convex, $H_A^* = I_A$. + \end{enumerate} +\end{lemma} +\begin{proof} + (1): Since $\ol{\conv}(A) \supset A$, $H_A \le H_{\ol{\conv}(A)}$. On the other hand, for each $\phi \in F$, $A \subset \bracs{\phi \le H_A(\phi)}$. Since $\bracs{\phi \le H_A(\phi)}$ is a $\sigma(E, F)$-closed convex set, it contains $\ol{\conv}(A)$. Thus $\ol{\conv}(A) \subset \bracs{\phi \le H_A(\phi)}$ and $H_{\ol{\conv}(A)}(\phi) \le H_A(\phi)$. + + (2): Using (1), assume without loss of generality that $B$ is $\sigma(E, F)$-closed and convex. Suppose that $H_A \le H_B$, then $A \subset \bigcap_{\phi \in F}\bracs{\phi \le H_B(\phi)}$. By the \hyperref[Hahn-Banach Theorem]{theorem:hahn-banach-geometric-2}, $B = \bigcap_{\phi \in F}\bracs{\phi \le H_B(\phi)}$, so $A \subset B$. + + (3): Let $\phi \in F$, then since $I_A|_{A^c} = \infty$, + \begin{align*} + I_A^*(\phi) &= \sup_{x \in E}\dpn{x, \phi}{\lambda} - I_A(x) = \sup_{x \in A}\dpn{x, \phi}{\lambda} - I_A(x) \\ + &= \sup_{x \in A}\dpn{x, \phi}{\lambda} = H_A(\phi) + \end{align*} + + (4): Given that $A$ is $\sigma(E, F)$-closed and convex, $I_S$ is convex and lower semicontinuous. By the \hyperref[Fenchel-Moreau Theorem]{theorem:fenchel-moreau} and (3), $I_A = I_A^{**} = H_A^*$. +\end{proof} + +\begin{theorem} +\label{theorem:support-function-seminorm} + Let $\dpn{E, F}{\lambda}$ be a duality over $\real$, $f: X \to (-\infty, \infty]$ be a $\sigma(E, F)$-lower semicontinuous, subadditive, and positively homogeneous function with $f(0) = 0$, and + \[ + \Sigma = \bracs{\phi \in F| (\phi, 0) \le f} + \] + + then: + \begin{enumerate} + \item $f^* = I_\Sigma$. + \item $\Sigma$ is the unique non-empty $\sigma(F, E)$-closed convex subset of $F$ such that $f = H_\Sigma$. + \item $\Sigma$ is equicontinuous if and only if there exists $U \in \cn_E(0)$ such that $\sup_{x \in U}f(x) < \infty$. + \end{enumerate} + + Conversely, + \begin{enumerate}[start=2] + \item For any non-empty $\Sigma \subset A$, $H_\Sigma$ is a lower semicontinuous, subadditive, and positively homogeneous function with $H_\Sigma(0) = 0$. + \end{enumerate} +\end{theorem} +\begin{proof}[Proof, {{\cite[Theorem 4.25]{Clarke}}}. ] + (1): Let $\phi \in \Sigma$, then since $\dpn{x, \phi}{\lambda} \le f(x)$ for all $x \in E$, + \[ + 0 \ge f^*(\phi) = \sup_{x \in E}\dpn{x, \phi}{\lambda} - f(x) = \dpn{0, \phi}{\lambda} - f(0) = 0 + \] + + where the supremum is achieved at $0$. On the other hand, if $\phi \in F \setminus \Sigma$, then there exists $x \in E$ such that $\dpn{x, \phi}{\lambda} - f(x) > 0$. In which case, by positive homogeneity of $f$, + \[ + f^*(\phi) \ge \sup_{\mu > 0}\dpn{\mu x, \phi}{\lambda} - f(\mu x) = \sup_{\mu > 0}\mu\braks{\dpn{ x, \phi} - f(x)} = \infty + \] + + Thus $f^{*} = I_\Sigma$. + + (2): By \autoref{lemma:lsc-affine-minorant}, $I_\Sigma \ne \infty$, so $\Sigma \ne \emptyset$. Since + \[ + \Sigma = \bigcap_{x \in E}\bracs{\phi \in F| \dpn{x, \phi}{\lambda} \le f(x)} + \] + + is an intersection of $\sigma(F, E)$-closed and convex sets, it is also $\sigma(F, E)$-closed. + + Now, let $x, y \in E$ and $t \in [0, 1]$, then since $f$ is subadditive and positively homogeneous, + \[ + f((1 - t)x + ty) \le f((1 - t)x) + f(ty) = (1 - t)f(x) + tf(y) + \] + + so $f$ is convex. Given that $f$ is also $\sigma(E, F)$-lower semicontinuous, the \hyperref[Fenchel-Moreau Theorem]{theorem:fenchel-moreau} and (4) of \autoref{lemma:support-function-gymnastics} imply that + \[ + f = f^{**} = I_\Sigma^* = H_\Sigma + \] + + By (2) of \autoref{lemma:support-function-gymnastics}, $\Sigma$ is the unique closed $\sigma(F, E)$-convex set such that $f = H_\Sigma$. + + (3): Let $U \in \cn_E(0)$ be circled such that $M = \sup_{x \in U}f(x) < \infty$, then + \[ + \sup_{\substack{y \in \Sigma \\ x \in U}}\dpn{x, y}{\lambda} \le \sup_{x \in U}f(x) = M < \infty + \] + + Since $U$ is circled, $\bigcup_{y \in \Sigma}\dpn{M^{-1}U, y}{\lambda} \subset \ol{B_\real(0, 1)}$, so $\Sigma$ is equicontinuous by \autoref{proposition:equicontinuous-linear}. + + Conversely, if $\Sigma$ is equicontinuous, then there exists $U \in \cn_E(0)$ such that $M = \sup_{y \in \Sigma, x \in U}\dpn{x, y}{\lambda} < \infty$. In which case, (2) implies that + \[ + \sup_{x \in U}f(x) = \sup_{x \in U}H_\Sigma(x) = \sup_{\substack{y \in \Sigma \\ x \in U}} \dpn{x, y}{\lambda} = M < \infty + \] + + (4): By \autoref{proposition:semicontinuous-properties}, $H_\Sigma$ is lower semicontinuous. + + Let $x, y \in E$ and $\mu > 0$, then + \begin{align*} + H_\Sigma(x + y) &= \sup_{z \in \Sigma}\dpn{x + y, z}{\lambda} \\ + &\le \sup_{z \in \Sigma}\dpn{x, z}{\lambda} + \sup_{z \in \Sigma}\dpn{y, z}{\lambda} = H_\Sigma(x) + H_\Sigma(y) + \end{align*} + + and + \[ + H_\Sigma(\mu x) = \sup_{z \in \Sigma}\dpn{\mu x, z}{\lambda} = \mu\sup_{z \in \Sigma}\dpn{x, z}{\lambda} = \mu H_\Sigma(x) + \] + + so $H_\Sigma$ is subadditive and positively homogeneous. + + +\end{proof} + + + diff --git a/src/fa/duality/polar.tex b/src/fa/duality/polar.tex index 85b0c05..8c1c3e8 100644 --- a/src/fa/duality/polar.tex +++ b/src/fa/duality/polar.tex @@ -127,15 +127,6 @@ Therefore $y \in A^{\circ}$, but $\text{Re}\dpn{x_0, y}{\lambda} > 1$, so $x_0 \not\in A^{\circ\circ}$. \end{proof} -\begin{proposition} -\label{proposition:polar-equicontinuous} - Let $E$ be a locally convex space, then: - \begin{enumerate} - \item For each equicontinuous family $\cf \subset E^*$ equicontinuous, $\cf^\circ \in \cn_E(0)$. - \item For each $U \in \cn_E(0)$, $U^\circ$ is equicontinuous. - \end{enumerate} -\end{proposition} - diff --git a/src/fa/notation.tex b/src/fa/notation.tex index cea2534..26cbc5d 100644 --- a/src/fa/notation.tex +++ b/src/fa/notation.tex @@ -53,6 +53,8 @@ $\partial f(x)$ & Subdifferential of $f$ at $x$. & \autoref{definition:subgradient} \\ $(\phi, \alpha) \le f$ & $\phi - \alpha \le f$. $(\phi, \alpha)$ is an affine minorant of $f$. & \autoref{definition:affine-minorant} \\ $f^*$ & Conjugate function of $f$. & \autoref{definition:conjugate-function} \\ + $I_A$ & Indicator/infinity characteristic function of $A$. & \autoref{definition:infinity-characteristic-function} \\ + $H_A$ & Support function of $A$ &\autoref{definition:support-function} \\ % ---- Interpolation Spaces ---- \\ $\catc_1$ & Category of compatible couples in $\catc$. & \autoref{definition:compatible-category} \\ $\cf(E_0, E_1)$ & Calderón space of $(E_0, E_1)$ & \autoref{definition:calderon-space} \\ diff --git a/src/fa/tvs/equicontinuous.tex b/src/fa/tvs/equicontinuous.tex index 998fc9d..2aef80e 100644 --- a/src/fa/tvs/equicontinuous.tex +++ b/src/fa/tvs/equicontinuous.tex @@ -1,7 +1,7 @@ \section{Equicontinuous Families of Linear Maps} \label{section:equicontinuous-linear} -\begin{proposition}[{{\cite[IV.4.2]{SchaeferWolff}}}] +\begin{proposition} \label{proposition:equicontinuous-linear} Let $E, F$ be TVSs over $K \in RC$ and $\alg \subset \hom(E; F)$, then the following are equivalent: \begin{enumerate} @@ -12,7 +12,7 @@ \item For each $V \in \cn_F(0)$, $\bigcap_{T \in \alg}T^{-1}(V) \in \cn_E(0)$. \end{enumerate} \end{proposition} -\begin{proof} +\begin{proof}[Proof, {{\cite[IV.4.2]{SchaeferWolff}}}. ] (5) $\Rightarrow$ (1): Let $V \in \cn_F(0)$, then $U = \bigcap_{T \in \alg}T^{-1}(V) \in \cn_E(0)$. Thus for any $x, y \in E$ with $x - y \in U$, $Tx - Ty \in V$ for all $T \in \alg$. \end{proof}