Editing Weak topology (section)

=== Weak convergence ===
{{further|Weak convergence (Hilbert space)}}

The weak topology is characterized by the following condition: a [[net (mathematics)|net]] <math>(x_\lambda)</math> in {{mvar|X}} converges in the weak topology to the element {{mvar|x}} of {{mvar|X}} if and only if <math>\phi(x_\lambda)</math> converges to <math>\phi(x)</math> in <math>\mathbb{R}</math> or <math>\mathbb{C}</math> for all <math>\phi\in X^*</math>.

In particular, if <math>x_n</math> is a [[sequence (mathematics)|sequence]] in {{mvar|X}}, then <math>x_n</math> '''converges weakly to''' {{mvar|x}} if

:<math>\varphi(x_n) \to \varphi(x)</math>

as {{math|''n'' → ∞}} for all <math>\varphi \in X^*</math>. In this case, it is customary to write

:<math>x_n \overset{\mathrm{w}}{\longrightarrow} x</math>

or, sometimes,

:<math>x_n \rightharpoonup x.</math>