functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...

, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the

locally convex In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

on the set of

bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vecto ...

s on a

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...

''H'' induced by the seminorms of the form

T\mapsto\, Tx\,

, as ''x'' varies in ''H''. Equivalently, it is the

coarsest topology In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as th ...

such that, for each fixed ''x'' in ''H'', the evaluation map

T\mapsto Tx

(taking values in ''H'') is continuous in T. The equivalence of these two definitions can be seen by observing that a

subbase In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by s ...

for both topologies is given by the sets

U(T_0,x,\epsilon) = \

(where ''T₀'' is any bounded operator on ''H'', ''x'' is any vector and ε is any positive real number). In concrete terms, this means that

T_i\to T

in the strong operator topology if and only if

\, T_ix-Tx\, \to 0

for each ''x'' in ''H''. The SOT is stronger than the

weak operator topology In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number \langle Tx, y\rangle is ...

and weaker than the

norm topology In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Intro ...

. The SOT lacks some of the nicer properties that the

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 In mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra. Theorem Theorem. Let ' ...

. The

linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , th ...

s on the set of bounded operators on a Hilbert space that are continuous in the SOT are precisely those continuous in the

(WOT). Because of this, the closure of a

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...

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.

References

* * * * * {{Duality and spaces of linear maps Banach spaces Topology of function spaces

See also

References