Editing Weak topology (section)

==Operator topologies==

If {{mvar|X}} and {{mvar|''Y''}} are topological vector spaces, the space {{math|''L''(''X'',''Y'')}} of [[continuous linear operator]]s {{math|''f'' : ''X''&nbsp;→&nbsp;''Y''}} may carry a variety of different possible topologies. The naming of such topologies depends on the kind of topology one is using on the target space {{mvar|''Y''}} to define operator convergence {{harv|Yosida|1980|loc=IV.7 Topologies of linear maps}}. There are, in general, a vast array of possible [[operator topology|operator topologies]] on {{math|''L''(''X'',''Y'')}}, whose naming is not entirely intuitive.

For example, the '''[[strong operator topology]]''' on {{math|''L''(''X'',''Y'')}} is the topology of ''pointwise convergence''. For instance, if {{mvar|''Y''}} is a normed space, then this topology is defined by the seminorms indexed by {{math|''x'' &isin; ''X''}}:

:<math>f\mapsto \|f(x)\|_Y.</math>

More generally, if a family of seminorms ''Q'' defines the topology on {{mvar|''Y''}}, then the seminorms {{math|''p''<sub>''q'', ''x''</sub>}} on {{math|''L''(''X'',''Y'')}} defining the strong topology are given by

:<math>p_{q,x} : f \mapsto q(f(x)),</math>

indexed by {{math|''q'' &isin; ''Q''}} and {{math|''x'' &isin; ''X''}}.

In particular, see the [[weak operator topology]] and [[weak* operator topology]].