My screencast on the Uniform Boundedness Principle (which some call the Banach-Steinhaus Theorem) is available from
This is one of the screencasts (video with synchronized audio) from my level 4 module G14FUN Functional Analysis. (See also the complete set of G!4FUN screencasts available from http://wp.me/PosHB-8v )
There is some inconsistency in the literature over which theorem should be called the Banach-Steinhaus Theorem. In this screencast I cover two results, one of which is a corollary of the other. Both of these results have been called the Banach-Steinhaus Theorem (by various authors). The stronger of the two results is the one which is also known as the Uniform Boundedness Principle.