Operator topologies

In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B ( X ) {displaystyle B(X)} of bounded linear operators on a Banach space X {displaystyle X} . In the mathematical field of functional analysis there are several standard topologies which are given to the algebra B ( X ) {displaystyle B(X)} of bounded linear operators on a Banach space X {displaystyle X} . Let ( T n ) n ∈ N {displaystyle (T_{n})_{nin mathbb {N} }} be a sequence of linear operators on the Banach space X {displaystyle X} . Consider the statement that ( T n ) n ∈ N {displaystyle (T_{n})_{nin mathbb {N} }} converges to some operator T {displaystyle T} on X {displaystyle X} . This could have several different meanings: There are many topologies that can be defined on B ( X ) {displaystyle B(X)} besides the ones used above; most are at first only defined when X = H {displaystyle X=H} is a Hilbert space, even though in many cases there are appropriate generalisations. The topologies listed below are all locally convex, which implies that they are defined by a family of seminorms.