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In
functional analysis Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...
, a branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, 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 spa ...
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
on the set of
bounded operator In functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis ...
s on a
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ''x'' varies in ''H''. Equivalently, it is the
coarsest topologyIn 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 the c ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...
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 Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in 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 (mathematics), norm defined on the space of bounded linear operators between two given normed vect ...
. The SOT lacks some of the nicer properties that the
weak operator topology In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional ...
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 In functional analysis, a branch of mathematics, the Borel functional calculus is a ''functional calculus'' (that is, an assignment of operator (mathematics), operators from commutative algebras to functions defined on their Spectrum of a ring, spec ...
, just as the norm topology does for the
continuous functional calculus In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
. The
linear functional In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional ...
(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 Real number, 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 o ...

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 In mathematics, a ''C''0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient ordinary differen ...
*
Topologies on the set of operators on a Hilbert space In the mathematics, mathematical field of functional analysis there are several standard topology, topologies which are given to the algebra of bounded linear operators on a Banach space . Introduction Let (T_n)_ be a sequence of linear operato ...


References

* * * * * {{DualityInLCTVSs Topology of function spaces