View source for Strong operator topology

{{Short description|Locally convex topology on function spaces}}
In [[functional analysis]], a branch of [[mathematics]], the '''strong operator topology''', often abbreviated '''SOT''', is the [[Locally convex topological vector space|locally convex]] [[topology]] on the set of [[bounded operator]]s on a [[Hilbert space]] ''H'' induced by the [[Seminorm|seminorms]] of the form <math>T\mapsto\|Tx\|</math>, as ''x'' varies in ''H''.

Equivalently, it is the [[Initial topology|coarsest topology]] such that, for each fixed ''x'' in ''H'', the evaluation map <math>T\mapsto Tx</math> (taking values in ''H'') is continuous in T. The equivalence of these two definitions can be seen by observing that a [[subbase]] for both topologies is given by the sets <math>U(T_0,x,\epsilon) = \{T : \|Tx-T_0x\| < \epsilon\}</math> (where ''T<sub>0</sub>'' is any bounded operator on ''H'', ''x'' is any vector and ε is any positive real number).

In concrete terms, this means that <math>T_i\to T</math> in the strong operator topology if and only if <math>\|T_ix-Tx\|\to 0</math> for each ''x'' in ''H''.

The SOT is [[Finer topology|stronger]] than the [[weak operator topology]] and weaker than the [[Operator norm|norm topology]].

The SOT lacks some of the nicer properties that the [[weak operator topology]] has, but being stronger, things are sometimes easier to prove in this topology. It can be viewed as more natural, too, since it is simply the topology of pointwise convergence.

The SOT topology also provides the framework for the [[measurable functional calculus]], just as the norm topology does for the [[continuous functional calculus]]. 

The [[linear functional]]s on the set of bounded operators on a Hilbert space that are continuous in the SOT are precisely those continuous in the [[weak operator topology]] (WOT). Because of this, the closure of a [[convex set]] of operators in the WOT is the same as the closure of that set in the SOT.

This language translates into convergence properties of Hilbert space operators. For a complex Hilbert space, it is easy to verify by the polarization identity, that Strong Operator convergence implies Weak Operator convergence.

==See also==

* [[Strongly continuous semigroup]]
* [[Topologies on the set of operators on a Hilbert space]]

== References ==

{{reflist}}

* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn | Rudin | 1991 | p=}} -->
* {{cite book|last=Pedersen|first=Gert|title=Analysis Now|year=1989|publisher=Springer|isbn=0-387-96788-5}}
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | Wolff | 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Trèves | 2006 | p=}} -->

{{Banach spaces}}
{{Hilbert space}}
{{Functional analysis}}
{{Duality and spaces of linear maps}}

[[Category:Banach spaces]]
[[Category:Topology of function spaces]]