Editing Operator topologies (section)

== Introduction ==

Let <math>(T_n)_{n \in \mathbb N}</math> be a sequence of linear operators on the Banach space <math>X</math>. Consider the statement that <math>(T_n)_{n \in \N}</math> converges to some operator <math>T</math> on <math>X</math>. 
This could have several different meanings:
* If <math>\|T_n - T\| \to 0</math>, that is, the [[operator norm]] of <math>T_n - T</math> (the supremum of <math>\| T_n x - T x \|_X</math>, where <math>x</math> ranges over the [[unit ball]] in <math>X</math>) converges to <math>0</math>, we say that <math>T_n \to T</math> in the '''[[uniform operator topology]]'''.
* If <math>T_n x \to Tx</math> for all <math>x \in X</math>, then we say <math>T_n \to T</math> in the '''[[strong operator topology]]'''.
* Finally, suppose that for all <math>x \in X</math> we have <math>T_n x \to Tx</math> in the [[weak topology]] of <math>X</math>. This means that <math>F(T_n x) \to F(T x)</math> for all continuous linear functionals <math>F</math> on <math>X</math>. In this case we say that <math>T_n \to T</math> in the '''[[weak operator topology]]'''.