Weakly Compact Set in Banach Space is Angelic

Theorem

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,}_X}$ be a Banach space over $\GF$.

Let $w$ be the weak topology on $\struct {X, \norm {\, \cdot \,}_X}$.

Let $\struct {K, w}$ be compact.

Then $\struct {K, w}$ is angelic.

Let $\cl_w$ be the closure taken in $\struct {X, w}$.

Let $\cl$ be the closure taken in $\struct {X, \norm {\, \cdot \,}_X}$.

Let $A \subseteq K$.

Let $x \in \map {\cl_w} A$.

Hence there exists a countable set $S \subseteq A$ such that $x \in \map {\cl_w} S$.

Let:

$L = \map \cl {\map \span S}$

From Topological Closure is Closed, $L$ is closed in $\struct {X, \norm {\, \cdot \,}_X}$.

From Closed Linear Span is Closed Vector Subspace, $L$ is a norm-closed linear subspace of $X$.

Hence we have $\map {\cl_w} S \subseteq L$.

Further, since $S \subseteq A$, we have $\map {\cl_w} S \subseteq \map {\cl_w} A \subseteq K$.

Hence $\map {\cl_w} S$ is closed in $\struct {K, w}$.

Since $\struct {K, w}$ is compact, we have that $\struct {\map {\cl_w} S, w}$ is compact from Closed Subspace of Compact Space is Compact.

Since $L$ is separable, we obtain that $\struct {\map {\cl_w} S, w}$ is metrizable from Weakly Compact Set in Separable Banach Space is Weakly Metrizable.

Recall that $x \in \map {\cl_w} S$.

Hence, from Point in Closure of Subset of Metric Space iff Limit of Sequence, there exists a sequence in $S$ such that $x_n \to x$ in $\struct {\map {\cl_w} S, w}$.

Since $S \subseteq A$, we have obtained a sequence in $A$ converging weakly to $x$.

Since $x \in \map {\cl_w} A$ was arbitrary, we have that $\struct {K, w}$ is angelic.

$\blacksquare$

2001: Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant and Václav Zizler: Functional Analysis and Infinite-Dimensional Geometry ... (previous) ... (next): Theorem $4.50$