pointwise limit of bounded operators is bounded

The following result is a corollary of the principle of uniform boundednessMathworldPlanetmath.

Theorem - Let X be a Banach spaceMathworldPlanetmath and Y a normed vector spacePlanetmathPlanetmath. Let (Tn)B(X,Y) be a sequence of bounded operatorsMathworldPlanetmathPlanetmath from X to Y. If (Tnx) converges for every xX, then the operator



is linear and . Moreover, the sequence (Tn) is bounded (http://planetmath.org/Bounded).

Proof : It is clear that the operator T is linear.

For each xX we have supnTnx< since (Tnx) is . By the principle of uniform boundedness (http://planetmath.org/BanachSteinhausTheorem) we must also have M:=supnTn<.

Then for each xX we have


which means that T is .

Title pointwise limit of bounded operators is bounded
Canonical name PointwiseLimitOfBoundedOperatorsIsBounded
Date of creation 2013-03-22 17:32:10
Last modified on 2013-03-22 17:32:10
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 4
Author asteroid (17536)
Entry type Corollary
Classification msc 46B99
Classification msc 47A05