proof of open mapping theorem

We prove that if Λ:XY is a continuousMathworldPlanetmath linear surjective map between Banach spacesMathworldPlanetmath, then Λ is an open map. It suffices to show that Λ maps the open unit ball in X to a neighborhood of the origin of Y.

Let U, V be the open unit balls in X, Y respectively. Then X=kkU, so, since Λ is surjective, Y=Λ(X)=Λ(kkU)=kΛ(kU). By the Baire category theorem, Y is not the union of countably many nowhere dense sets, so there is some k and some open set WY such that W is contained in the closurePlanetmathPlanetmath of Λ(kU).

Let y0W, and pick η>0 so that y0+yW for all y with ||y||<η. Then y0 and y0+y are limit pointsMathworldPlanetmathPlanetmath of Λ(kU), so there are sequences xi and xi′′ in kU with Λxiy0 and Λxi′′y0+y. Letting xi=xi′′-xi, we have ||xi||<2k and Λxiy. So for any yηV there is a sequence xi in 2kU with Λxiy. Then by the linearity of Λ, we have that for any ϵ>0 and any yY, there is an xX with:

||x||<δ-1||y|| and ||Λx-y||<ϵ (1)

where δ=η/2k.

Now let yδV and ϵ>0. Then there is some x1 with ||x1||<1 and ||y-Λx1||<ϵδ. Define a sequence xn inductively as follows. Assume:

||y-Λ(x1+x2++xn)||<ϵδ2-n (2)

Then by (1) we can pick xn+1 so that:

||xn+1||<ϵ2-n (3)

and ||y-Λ(x1+x2++xn)-Λ(xn+1)||<ϵδ2-(n+1), so (2) is satisfied for xn+1.

Put sn=x1+x2++xn. Then from (3), sn is a Cauchy sequencePlanetmathPlanetmath, and so, since X is complete, it converges to some xX. By (2), Λsny, and by the continuity of Λ, ΛsnΛx, so Λx=y. Also, ||x||=limn||sn||n=1||xn||<1+ϵ. Thus Λ((1+ϵ)U)ηV, or Λ(U)(1+ϵ)-1δV. Since this is true for all ϵ>0, we have Λ(U)ϵ>0(1+ϵ)-1δV=δV.

Title proof of open mapping theorem
Canonical name ProofOfOpenMappingTheorem
Date of creation 2013-03-22 16:23:31
Last modified on 2013-03-22 16:23:31
Owner Statusx (15142)
Last modified by Statusx (15142)
Numerical id 9
Author Statusx (15142)
Entry type Proof
Classification msc 30A99
Classification msc 46A30