A Hilbert Space Problem Book pp 59-62 | Cite as

# Operator Topologies

Chapter

## Abstract

**107. Topologies for operators**. A Hilbert space has two useful topologies (weak and strong); the space of operators on a Hilbert space has several. The metric topology induced by the norm is one of them; to distinguish it from the others, it is usually called the

*norm topology*or the

*uniform topology*. The next two are natural outgrowths for operators of the strong and weak topologies for vectors. A subbase for the

*strong*operator topology is the collection of all sets of the form

$$\left\{ {A:\,\parallel \,\left( {A\, - \,{A_0}} \right)f\parallel \, < \,\varepsilon } \right\};$$

$$\left\{ {A:\,\,\parallel \,\left( {A\, - \,{A_0}} \right){f_i}\,\parallel \,\, < \,\varepsilon ,\,\,i\, = \,1,\,...\,,\,k} \right\}.$$

*k*is a positive integer,

*f*

_{ 1 }… ,

*f*

_{ k }are vectors, and

*ε*is a positive number. A subbase for the

*weak*operator topology is the collection of all sets of the form

$$\left\{ {A:\left| {\left( {\left( {A\, - \,{A_0}} \right)\,f,\,g} \right)} \right|\, < \,\varepsilon } \right\},$$

*f*and

*g*are vectors and

*ε >*0; as above (as always) a base is the collection of all finite intersections of such sets. The corresponding concepts of convergence (for sequences and nets) are easy to describe:

*A*

_{ n }→

*A*strongly if and only if

*A*

_{ n }

*f*→

*Af*strongly for each

*f*(i.e., ||

*(A*

_{ n }–

*A)f||*→ 0 for each

*f*), and

*A*

_{ n }→

*A*weakly if and only if

*A*

_{ n }

*f*→

*Af*weakly for each

*f*(i.e.,(A

_{ n }

*f, g)*→

*(Af, g)*for each

*f*and

*g)*. For a slightly different and often very efficient definition of the strong and weak operator topologies see Problems 224 and 225.

## Keywords

Hilbert Space Weak Topology Norm Topology Sequential Continuity Strong Topology
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag New York Inc. 1982