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## Preview

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## Copyright information

© Springer-Verlag New York Inc. 1982