User:Caliburn/s/fa/Helly's Theorem

Theorem

Let $\struct {X, \norm {\, \cdot \,}_X}$ be a separable normed vector space.

Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,}_X}$.

Let $\sequence {f_n}_{n \mathop \in \N}$ be a bounded sequence in $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$

Then $\sequence {f_n}_{n \mathop \in \N}$ has a subsequence $\sequence {f_{n_j} }_{j \mathop \in \N}$ that is weakly-$\ast$ convergent.

Proof

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Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $27.5$: Two Weak-Compactness Theorems