Editing Banach space (section)

====Sequences, weak and weak* compactness====

When <math>X</math> is separable, the unit ball of the dual is weak*-compact by the [[Banach–Alaoglu theorem]] and metrizable for the weak* topology,<ref name="DualBall" /> 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 <math>X</math> is metrizable if and only if <math>X</math> is finite-dimensional.<ref>see Proposition&nbsp;2.5.14, p.&nbsp;215 in {{harvtxt|Megginson|1998}}.</ref> If the dual <math>X'</math> is separable, the weak topology of the unit ball of <math>X</math> 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.

{{math theorem| name = [[Eberlein–Šmulian theorem]]<ref>see for example p.&nbsp;49, II.C.3 in {{harvtxt|Wojtaszczyk|1991}}.</ref> | math_statement = A set <math>A</math> in a Banach space is relatively weakly compact if and only if every sequence <math>\{a_n\}</math> in <math>A</math> has a weakly convergent subsequence.}}

A Banach space <math>X</math> is reflexive if and only if each bounded sequence in <math>X</math> has a weakly convergent subsequence.<ref>see Corollary&nbsp;2.8.9, p.&nbsp;251 in {{harvtxt|Megginson|1998}}.</ref>

A weakly compact subset <math>A</math> in <math>\ell^1</math> is norm-compact. Indeed, every sequence in <math>A</math> has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of <math>\ell^1.</math>