Weak operator topology

In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, the weak operator topology, often abbreviated WOT, is the weakest <a href="/facts/Topology/2EqW47cU">topology</a> on the set of <a href="/facts/Bounded_operator/LYmDTIqR">bounded operators</a> on a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> 
 
 
 
 H
 
 
 {\displaystyle H}
 
, such that the <a href="/facts/Functional_(mathematics)/3Cp3PHBR">functional</a> sending an operator 
 
 
 
 T
 
 
 {\displaystyle T}
 
 to the complex number 
 
 
 
 ⟨
 T
 x
 ,
 y
 ⟩
 
 
 {\displaystyle \langle Tx,y\rangle }
 
 is <a href="/facts/Continuous_function/sKbl02pB">continuous</a> for any vectors 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 y
 
 
 {\displaystyle y}
 
 in the Hilbert space.
Explicitly, for an operator 
 
 
 
 T
 
 
 {\displaystyle T}
 
 there is <a href="/facts/Neighborhood_basis/fMRM8Xht">base of neighborhoods</a> of the following type: choose a finite number of vectors 
 
 
 
 
 x
 
 i
 
 
 
 
 {\displaystyle x_{i}}
 
, continuous functionals 
 
 
 
 
 y
 
 i
 
 
 
 
 {\displaystyle y_{i}}
 
, and positive real constants 
 
 
 
 
 ε
 
 i
 
 
 
 
 {\displaystyle \varepsilon _{i}}
 
 indexed by the same finite set 
 
 
 
 I
 
 
 {\displaystyle I}
 
. An operator 
 
 
 
 S
 
 
 {\displaystyle S}
 
 lies in the neighborhood if and only if 
 
 
 
 
 |
 
 
 y
 
 i
 
 
 (
 T
 (
 
 x
 
 i
 
 
 )
 −
 S
 (
 
 x
 
 i
 
 
 )
 )
 
 |
 
 <
 
 ε
 
 i
 
 
 
 
 {\displaystyle |y_{i}(T(x_{i})-S(x_{i}))|<\varepsilon _{i}}
 
 for all 
 
 
 
 i
 ∈
 I
 
 
 {\displaystyle i\in I}
 
.
Equivalently, a <a href="/facts/Net_(mathematics)/msmsvMTT">net</a> 
 
 
 
 
 T
 
 i
 
 
 ⊆
 B
 (
 H
 )
 
 
 {\displaystyle T_{i}\subseteq B(H)}
 
 of bounded operators converges to 
 
 
 
 T
 ∈
 B
 (
 H
 )
 
 
 {\displaystyle T\in B(H)}
 
 in WOT if for all 
 
 
 
 y
 ∈
 
 H
 
 ∗
 
 
 
 
 {\displaystyle y\in H^{*}}
 
 and 
 
 
 
 x
 ∈
 H
 
 
 {\displaystyle x\in H}
 
, the net 
 
 
 
 y
 (
 
 T
 
 i
 
 
 x
 )
 
 
 {\displaystyle y(T_{i}x)}
 
 converges to 
 
 
 
 y
 (
 T
 x
 )
 
 
 {\displaystyle y(Tx)}
 
.

Weak operator topology open-in-new

Weak operator topology