Weak operator topology

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In functional analysis, the weak operator topology, often abbreviated WOT,^[1] is the weakest topology on the set of bounded operators on a Hilbert space ${\displaystyle H}$, such that the functional sending an operator ${\displaystyle T}$ to the complex number ${\displaystyle \langle Tx,y\rangle }$ is continuous for any vectors ${\displaystyle x}$ and ${\displaystyle y}$ in the Hilbert space.

Explicitly, for an operator ${\displaystyle T}$ there is base of neighborhoods of the following type: choose a finite number of vectors ${\displaystyle x_{i}}$, continuous functionals ${\displaystyle y_{i}}$, and positive real constants ${\displaystyle \varepsilon _{i}}$ indexed by the same finite set ${\displaystyle I}$. An operator ${\displaystyle S}$ lies in the neighborhood if and only if ${\displaystyle |y_{i}(T(x_{i})-S(x_{i}))|<\varepsilon _{i}}$ for all ${\displaystyle i\in I}$.

Equivalently, a net ${\displaystyle T_{i}\subseteq B(H)}$ of bounded operators converges to ${\displaystyle T\in B(H)}$ in WOT if for all ${\displaystyle y\in H^{*}}$ and ${\displaystyle x\in H}$, the net ${\displaystyle y(T_{i}x)}$ converges to ${\displaystyle y(Tx)}$.