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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 vec ...

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

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 vector ...

s on a

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

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

T\mapsto\, Tx\,

, as ''x'' varies in ''H''. Equivalently, it is the coarsest topology 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 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,Ilijas Farah, Combinatorial Set Theory of C*-algebras' (2019), p. 80. is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional ...

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. Informal ...

. 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, the continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra. In advanced theory, the ap ...

. The

linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...

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 set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...

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