Editing Weak topology (section)

== Weak-* topology ==
<!-- weak* convergence in normed linear space links to this heading -->
{{See also|Polar topology}}

The weak* topology is an important example of a [[polar topology]].

A space {{mvar|X}} can be embedded into its [[double dual]] ''X**'' by

:<math>x \mapsto \begin{cases} T_x: X^* \to \mathbb{K} \\ T_x(\phi) = \phi(x) \end{cases}</math>

Thus <math>T:X\to X^{**}</math> is an [[injective]] linear mapping, though not necessarily [[surjective]] (spaces for which ''this'' canonical embedding is surjective are called [[reflexive space|reflexive]]). The '''weak-* topology''' on <math>X^*</math> is the weak topology induced by the image of <math>T:T(X)\subset X^{**}</math>. In other words, it is the coarsest topology such that the maps ''T<sub>x</sub>'', defined by <math>T_x(\phi)=\phi(x)</math> from <math>X^*</math> to the base field <math>\mathbb{R}</math> or <math>\mathbb{C}</math> remain continuous.

;Weak-* convergence

A [[net (mathematics)|net]] <math>\phi_{\lambda}</math> in <math>X^*</math> is convergent to <math>\phi</math> in the weak-* topology if it converges pointwise:

:<math>\phi_{\lambda} (x) \to \phi (x)</math>

for all <math>x\in X</math>. In particular, a [[sequence (mathematics)|sequence]] of <math>\phi_n\in X^*</math> converges to <math>\phi</math> provided that

:<math>\phi_n(x)\to\phi(x)</math>

for all {{math|''x'' &isin; ''X''}}. In this case, one writes

:<math>\phi_n \overset{w^*}{\to} \phi</math>

as {{math|''n'' → ∞}}.

Weak-* convergence is sometimes called the '''simple convergence''' or the '''pointwise convergence'''. Indeed, it coincides with the [[pointwise convergence]] of linear functionals.

=== Properties ===

If {{mvar|X}} is a [[Separable space|separable]] (i.e. has a countable dense subset) [[locally convex]] space and ''H'' is a norm-bounded subset of its continuous dual space, then ''H'' endowed with the weak* (subspace) topology is a [[metrizable]] topological space.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}} However, for infinite-dimensional spaces, the metric cannot be translation-invariant.{{sfn | Folland | 1999 | pp=170}} If {{mvar|X}} is a separable [[Metrizable TVS|metrizable]] [[locally convex]] space then the weak* topology on the continuous dual space of {{mvar|X}} is separable.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}}

;Properties on normed spaces

By definition, the weak* topology is weaker than the weak topology on <math>X^*</math>. An important fact about the weak* topology is the [[Banach–Alaoglu theorem]]: if {{mvar|X}} is normed, then the closed unit ball in <math>X^*</math> is weak*-[[compact space|compact]] (more generally, the [[polar set|polar]] in <math>X^*</math> of a neighborhood of 0 in {{mvar|X}} is weak*-compact). Moreover, the closed unit ball in a normed space {{mvar|X}} is compact in the weak topology if and only if {{mvar|X}} is [[reflexive space|reflexive]].

In more generality, let {{mvar|F}} be locally compact valued field (e.g., the reals, the complex numbers, or any of the p-adic number systems). Let {{mvar|X}} be a normed topological vector space over {{mvar|F}}, compatible with the absolute value in {{mvar|F}}. Then in <math>X^*</math>, the topological dual space {{mvar|X}} of continuous {{mvar|F}}-valued linear functionals on {{mvar|X}}, all norm-closed balls are compact in the weak* topology.

If {{mvar|X}} is a normed space, a version of the [[Heine-Borel theorem]] holds. In particular, a subset of the continuous dual is weak* compact if and only if it is weak* closed and norm-bounded.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}} This implies, in particular, that when {{mvar|X}} is an infinite-dimensional normed space then the closed unit ball at the origin in the dual space of {{mvar|X}} does not contain any weak* neighborhood of 0 (since any such neighborhood is norm-unbounded).{{sfn | Narici | Beckenstein | 2011 | pp=225-273}} Thus, even though norm-closed balls are compact, X* is not weak* [[locally compact space|locally compact]].

If {{mvar|X}} is a normed space, then {{mvar|X}} is separable if and only if the weak* topology on the closed unit ball of <math>X^*</math> is metrizable,{{sfn | Narici | Beckenstein | 2011 | pp=225-273}} in which case the weak* topology is metrizable on norm-bounded subsets of <math>X^*</math>. If a normed space {{mvar|X}} has a dual space that is separable (with respect to the dual-norm topology) then {{mvar|X}} is necessarily separable.{{sfn | Narici | Beckenstein | 2011 | pp=225-273}} If {{mvar|X}} is a [[Banach space]], the weak* topology is not metrizable on all of <math>X^*</math> unless {{mvar|X}} is finite-dimensional.<ref>Proposition 2.6.12, p.&nbsp;226 in {{citation | last = Megginson | first = Robert E. | author-link = Robert Megginson | title = An introduction to Banach space theory | series = Graduate Texts in Mathematics | volume = 183 | publisher = Springer-Verlag | location = New York | year = 1998 | pages = xx+596 | isbn = 0-387-98431-3}}.</ref>