In mathematics, the requirements of functional analysis mean there are several standard topologies which are given to the set of bounded linear operators on a Hilbert space. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...

• norm topology
• ultrastrong topology
• strong operator topology (SOT)
• weak operator topology (WOT)
• weak-star operator topology
• ultraweak topology

In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... In functional analysis, the strong operator topology, often abbreviated SOT, is the weakest topology on the set of bounded operators on a Hilbert space such that the evaluation map sending an operator T to the real number is continuous for each vector x in the Hilbert space. ... In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space such that the functional sending an operator T to the complex number is continuous for any vectors x and y in the Hilbert space. ... In functional analysis, the weak-* topology on the set B(H) of bounded operators on a Hilbert space is the weak-* topology obtained from the predual of B(H), the trace class operators on H. See also: Topologies on the set of operators on a Hilbert space Categories: Operator theory...

Some facts

• The norm topology is stronger than the strong operator topology which is stronger than the weak operator topology.
• The norm topology is stronger than the weak-star topology which is stronger than the weak operator topology.
• The closures of a convex set in the strong and the weak operator topologies coincide.
• The weak operator topology and the weak-star topology agree on norm-bounded sets.

In mathematics, the possible topologies on a given set X form a partially ordered set: if a collection τ1 of subsets of X contains each subset in a collection τ2, and these are both topologies on X, we say that τ1 is a finer (alt. ... In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ...

See also

• Topology
• Hilbert space
• Bounded operator