Banach-Steinhaus theorem

Let X be a Banach spaceMathworldPlanetmath and Y a normed spaceMathworldPlanetmath. If a family (X,Y) of bounded operatorsMathworldPlanetmathPlanetmath from X to Y satisfies


for each xX, then


i.e. is a boundedPlanetmathPlanetmathPlanetmathPlanetmath subset of (X,Y) with the usual operator norm. In other words, there exists a constant c such that for all xX and T,

Title Banach-Steinhaus theorem
Canonical name BanachSteinhausTheorem
Date of creation 2013-03-22 14:48:39
Last modified on 2013-03-22 14:48:39
Owner Koro (127)
Last modified by Koro (127)
Numerical id 5
Author Koro (127)
Entry type Theorem
Classification msc 46B99
Synonym Principle of Uniform Boundedness
Synonym Uniform Boundedness Principle