Theorem 18.4.4.label 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
then:
- (1)
$f^{*} = I_{\Sigma}$.
- (2)
$\Sigma$ is the unique non-empty $\sigma(F, E)$-closed convex subset of $F$ such that $f = H_{\Sigma}$.
- (3)
$\Sigma$ is equicontinuous if and only if there exists $U \in \cn_{E}(0)$ such that $\sup_{x \in U}f(x) < \infty$.
Conversely,
- (3)
For any non-empty $\Sigma \subset A$, $H_{\Sigma}$ is a lower semicontinuous, subadditive, and positively homogeneous function with $H_{\Sigma}(0) = 0$.
Proof, [Theorem 4.25, Cla13]. (1): Let $\phi \in \Sigma$, then since $\dpn{x, \phi}{\lambda}\le f(x)$ for all $x \in E$,
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 $f$,
Thus $f^{*}= I_{\Sigma}$.
(2): By Lemma 18.3.7, $I_{\Sigma} \ne \infty$, so $\Sigma \ne \emptyset$. Since
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,
so $f$ is convex. Given that $f$ is also $\sigma(E, F)$-lower semicontinuous, the Fenchel-Moreau Theorem and (4) of Lemma 18.4.3 imply that
By (2) of Lemma 18.4.3, $\Sigma$ 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
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 Proposition 11.14.1.
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
(4): By Proposition 5.22.3, $H_{\Sigma}$ is lower semicontinuous.
Let $x, y \in E$ and $\mu > 0$, then
and
so $H_{\Sigma}$ is subadditive and positively homogeneous.$\square$
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