Editing Weak operator topology (section)

{{Short description|Weak topology on function spaces}}
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In [[functional analysis]], the '''weak operator topology''', often abbreviated '''WOT''',<ref>Ilijas Farah, ''[https://books.google.com/books?id=HtDGDwAAQBAJ&pg=PA80 Combinatorial Set Theory of C*-algebras]'' (2019), p. 80.</ref> is the weakest [[topology]] on the set of [[bounded operator]]s on a [[Hilbert space]] <math>H</math>, such that the [[functional (mathematics)|functional]] sending an operator <math>T</math> to the complex number <math>\langle Tx, y\rangle</math> is [[Continuous function|continuous]] for any vectors <math>x</math> and <math>y</math> in the Hilbert space.

Explicitly, for an operator <math>T</math> there is [[neighborhood basis|base of neighborhoods]] of the following type: choose a finite number of vectors <math>x_i</math>, continuous functionals <math>y_i</math>, and positive real constants <math>\varepsilon_i</math> indexed by the same finite set <math>I</math>. An operator <math>S</math> lies in the neighborhood if and only if <math>| y_i(T(x_i) - S(x_i))| < \varepsilon_i</math> for all <math>i \in I</math>.

Equivalently, a [[Net (mathematics)|net]] <math>T_i \subseteq B(H)</math> of bounded operators converges to <math>T \in B(H)</math> in WOT if for all <math> y \in H^*</math> and <math>x \in H</math>, the net <math>y(T_i x)</math> converges to <math> y(T x)</math>.