Weak operator topology
Weak topology on function spaces / From Wikipedia, the free encyclopedia
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 , such that the functional sending an operator
to the complex number
is continuous for any vectors
and
in the Hilbert space.
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Explicitly, for an operator there is base of neighborhoods of the following type: choose a finite number of vectors
, continuous functionals
, and positive real constants
indexed by the same finite set
. An operator
lies in the neighborhood if and only if
for all
.
Equivalently, a net of bounded operators converges to
in WOT if for all
and
, the net
converges to
.