proof of Banach-Steinhaus theorem


En={xX:T(x)n for all T}.

From the hypothesisMathworldPlanetmathPlanetmath, we have that


Also, each En is closed, since it can be written as


where B(0,n) is the closed ball centered at 0 with radius n in Y, and each of the sets in the intersectionMathworldPlanetmath is closed due to the continuity of the operators. Now since X is a Banach spaceMathworldPlanetmath, Baire’s category theoremMathworldPlanetmath implies that there exists n such that En has nonempty interior. So there is x0En and r>0 such that B(x0,r)En. Thus if xr, we have


for each T, and so


so if x1, we have


and this means that


for all T.

Title proof of Banach-Steinhaus theorem
Canonical name ProofOfBanachSteinhausTheorem
Date of creation 2013-03-22 14:48:41
Last modified on 2013-03-22 14:48:41
Owner Koro (127)
Last modified by Koro (127)
Numerical id 6
Author Koro (127)
Entry type Proof
Classification msc 46B99