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Sequences, weak and weak* compactness

When X {\displaystyle X} is separable, the unit ball of the dual is weak*-compact by the Banach–Alaoglu theorem and metrizable for the weak* topology, hence every bounded sequence in the dual has weakly* convergent subsequences. This applies to separable reflexive spaces, but more is true in this case, as stated below.

The weak topology of a Banach space X {\displaystyle X} is metrizable if and only if X {\displaystyle X} is finite-dimensional. If the dual X ′ {\displaystyle X'} is separable, the weak topology of the unit ball of X {\displaystyle X} is metrizable. This applies in particular to separable reflexive Banach spaces. Although the weak topology of the unit ball is not metrizable in general, one can characterize weak compactness using sequences.

Eberlein–Šmulian theorem — A set A {\displaystyle A} in a Banach space is relatively weakly compact if and only if every sequence { a n } {\displaystyle \left\{a_{n}\right\}} in A {\displaystyle A} has a weakly convergent subsequence.

A Banach space X {\displaystyle X} is reflexive if and only if each bounded sequence in X {\displaystyle X} has a weakly convergent subsequence.

A weakly compact subset A {\displaystyle A} in ℓ 1 {\displaystyle \ell ^{1}} is norm-compact. Indeed, every sequence in A {\displaystyle A} has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of ℓ 1. {\displaystyle \ell ^{1}.}