Editing Banach space (section)

===Weak convergences of sequences===

A sequence <math>\{x_n\}</math> in a Banach space <math>X</math> is ''weakly convergent'' to a vector <math>x \in X</math> if <math>\{f(x_n)\}</math> converges to <math>f(x)</math> for every continuous linear functional <math>f</math> in the dual <math>X'.</math> The sequence <math>\{x_n\}</math> is a ''weakly Cauchy sequence'' if <math>\{f(x_n)\}</math> converges to a scalar limit <math>L(f)</math> for every <math>f</math> in <math>X'.</math> 
A sequence <math>\{f_n\}</math> in the dual <math>X'</math> is ''weakly* convergent'' to a functional <math>f \in X'</math> if <math>f_n(x)</math> converges to <math>f(x)</math> for every <math>x</math> in <math>X.</math> 
Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the [[Uniform boundedness principle|Banach–Steinhaus]] theorem.

When the sequence <math>\{x_n\}</math> in <math>X</math> is a weakly Cauchy sequence, the limit <math>L</math> above defines a bounded linear functional on the dual <math>X',</math> that is, an element <math>L</math> of the bidual of <math>X,</math> and <math>L</math> is the limit of <math>\{x_n\}</math> in the weak*-topology of the bidual. 
The Banach space <math>X</math> is ''weakly sequentially complete'' if every weakly Cauchy sequence is weakly convergent in <math>X.</math> 
It follows from the preceding discussion that reflexive spaces are weakly sequentially complete.

{{math theorem| name = Theorem <ref>see III.C.14, p.&nbsp;140 in {{harvtxt|Wojtaszczyk|1991}}.</ref> | math_statement = For every measure <math>\mu,</math> the space <math>L^1(\mu)</math> is weakly sequentially complete.}}

An orthonormal sequence in a Hilbert space is a simple example of a weakly convergent sequence, with limit equal to the <math>\mathbf{0}</math> vector. 
The [[Schauder basis#Examples|unit vector basis]] of <math>\ell^p</math> for <math>1 < p < \infty,</math> or of <math>c_0,</math> is another example of a ''weakly null sequence'', that is, a sequence that converges weakly to <math>\mathbf{0}.</math> 
For every weakly null sequence in a Banach space, there exists a sequence of convex combinations of vectors from the given sequence that is norm-converging to <math>\mathbf{0}.</math><ref>see Corollary&nbsp;2, p.&nbsp;11 in {{harvtxt|Diestel|1984}}.</ref>

The unit vector basis of <math>\ell^1</math> is not weakly Cauchy. 
Weakly Cauchy sequences in <math>\ell^1</math> are weakly convergent, since <math>L^1</math>-spaces are weakly sequentially complete. 
Actually, weakly convergent sequences in <math>\ell^1</math> are norm convergent.<ref>see p.&nbsp;85 in {{harvtxt|Diestel|1984}}.</ref> This means that <math>\ell^1</math> satisfies [[Schur's property]].

====Results involving the {{math|𝓁<sup>1</sup>}} basis====

Weakly Cauchy sequences and the <math>\ell^1</math> basis are the opposite cases of the dichotomy established in the following deep result of&nbsp;H.&nbsp;P.&nbsp;Rosenthal.<ref>{{cite journal|last1=Rosenthal|first1=Haskell P|year=1974|title=A characterization of Banach spaces containing ℓ<sup>1</sup>|journal=Proc. Natl. Acad. Sci. U.S.A.|volume=71|issue=6| pages=2411–2413 | doi=10.1073/pnas.71.6.2411|pmid=16592162|pmc=388466|arxiv=math.FA/9210205|bibcode=1974PNAS...71.2411R|doi-access=free}} Rosenthal's proof is for real scalars. The complex version of the result is due to L. Dor, in {{cite journal| last1=Dor|first1=Leonard E|year=1975|title=On sequences spanning a complex ℓ<sup>1</sup> space|journal=Proc. Amer. Math. Soc. | volume=47|pages=515–516|doi=10.1090/s0002-9939-1975-0358308-x|doi-access=free}}</ref>

{{math theorem| name = Theorem<ref>see p.&nbsp;201 in {{harvtxt|Diestel|1984}}.</ref> | math_statement = Let <math>\{x_n\}_{n \in \N}</math> be a bounded sequence in a Banach space. Either <math>\{x_n\}_{n \in \N}</math> has a weakly Cauchy subsequence, or it admits a subsequence [[Schauder basis#Definitions|equivalent]] to the standard unit vector basis of <math>\ell^1.</math>}}

A complement to this result is due to Odell and Rosenthal&nbsp;(1975).

{{math theorem| name = Theorem<ref>{{citation|last1=Odell|first1=Edward W.|last2=Rosenthal|first2=Haskell P.|title=A double-dual characterization of separable Banach spaces containing ℓ<sup>1</sup>|journal=[[Israel Journal of Mathematics]]|volume=20|year=1975|issue=3–4 |pages=375–384|doi=10.1007/bf02760341|doi-access=free|s2cid=122391702|url=http://dml.cz/bitstream/handle/10338.dmlcz/133414/CommentatMathUnivCarolRetro_50-2009-1_5.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://dml.cz/bitstream/handle/10338.dmlcz/133414/CommentatMathUnivCarolRetro_50-2009-1_5.pdf |archive-date=2022-10-09 |url-status=live}}.</ref> | math_statement = Let <math>X</math> be a separable Banach space. The following are equivalent:
*The space <math>X</math> does not contain a closed subspace isomorphic to <math>\ell^1.</math>
*Every element of the bidual <math>X''</math> is the weak*-limit of a sequence <math>\{x_n\}</math> in <math>X.</math>}}

By the Goldstine theorem, every element of the unit ball <math>B''</math> of <math>X''</math> is weak*-limit of a net in the unit ball of <math>X.</math> When <math>X</math> does not contain <math>\ell^1,</math> every element of <math>B''</math> is weak*-limit of a {{em|sequence}} in the unit ball of <math>X.</math><ref>Odell and Rosenthal, Sublemma p.&nbsp;378 and Remark p.&nbsp;379.</ref>

When the Banach space <math>X</math> is separable, the unit ball of the dual <math>X',</math> equipped with the weak*-topology, is a metrizable compact space <math>K,</math><ref name="DualBall" /> and every element <math>x''</math> in the bidual <math>X''</math> defines a bounded function on <math>K</math>:
<math display=block>x' \in K \mapsto x''(x'), \quad |x''(x')| \leq \|x''\|.</math>

This function is continuous for the compact topology of <math>K</math> if and only if <math>x''</math> is actually in <math>X,</math> considered as subset of <math>X''.</math> 
Assume in addition for the rest of the paragraph that <math>X</math> does not contain <math>\ell^1.</math> 
By the preceding result of Odell and Rosenthal, the function <math>x''</math> is the [[Pointwise convergence|pointwise limit]] on <math>K</math> of a sequence <math>\{x_n\} \subseteq X</math> of continuous functions on <math>K,</math> it is therefore a [[Baire function|first Baire class function]] on <math>K.</math> 
The unit ball of the bidual is a pointwise compact subset of the first Baire class on <math>K.</math><ref>for more on pointwise compact subsets of the Baire class, see {{citation|last1=Bourgain|first1=Jean|author1-link=Jean Bourgain|last2=Fremlin|first2=D. H.|last3=Talagrand |first3=Michel|title=Pointwise Compact Sets of Baire-Measurable Functions|journal=Am. J. Math.|volume=100|year=1978|issue=4|pages=845–886|jstor=2373913|doi=10.2307/2373913}}.</ref>

====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>