The Minimization of Convex Functionals

Show that if a sequence of real numbers $\left\{ a_{n}\right\}$ converges to $a \in \overline{\mathbb {R}}$, then its sequence of arithmetic means also converges to $a$.

If $\Phi \in X^{*}$, prove that

For $1 < p < \infty$ define $C = \left\{ f \in L^{p}(E) \mid \left\| f\right\|_{p} \leq 1\right\}$. For $\Phi \in \left[L^{p}(E)\right]^{*}$, identify a function $f_{0} \in C$ at which the functional $\Phi : C \rightarrow \mathbb {R}$ has a minimum value.

Let $C = \left\{ f \in L^{1}[0,1] \mid \left\| f\right\|_{1} \leq 1\right\}$, and define $\Phi : C \rightarrow \mathbb {R}$ by $\Phi (f) = \int_{0}^{1} x f(x) d x$. Show that $C$ is a closed, bounded, convex subset of $L^{1}[0,1]$, that $\Phi : C \rightarrow \mathbb {R}$ is continuous and convex, but does not have a minimum value.

Let $C = \left\{ f \in C[0,1] \mid \left\| f\right\|_{\max } \leq 1\right\}$, and define $\Phi : C \rightarrow \mathbb {R}$ by $\Phi (f) = \int_{0}^{1 / 2} f(x) d x- \int_{1 / 2}^{1} f(x) d x$. Show that $C$ is a closed, bounded, convex subset of $C[0,1]$, that $\Phi : C \rightarrow \mathbb {R}$ is continuous and convex, but does not have a minimum value.

State and prove the Banach-Saks Theorem in $\ell^{2}$.

Let $m(E) < \infty$ and $p_{1}, p_{2} \in [1, \infty )$. Suppose $\varphi : \mathbb {R} \rightarrow \mathbb {R}$ is continuous and there are $c_{1}, c_{2} \geq 0$ for which $\left|\varphi (s)\right| \leq c_{1}+c_{2}\left|s\right|^{p_{1} / p_{2}}$ for all $s$. Show that

Let $C$ be a subset of a normed linear space $X$ with the property that each sequence in $C$ has a subsequence that converges strongly to a point in $C$. Show that $C$ is a closed and bounded subset of $X$.