bounded inverse theorem

The next result is a corollary of the open mapping theoremMathworldPlanetmath. It is often called the bounded inverse theorem or the inverse mapping theorem.

Theorem - Let X,Y be Banach spacesMathworldPlanetmath. Let T:XY be an invertible bounded operatorMathworldPlanetmathPlanetmath. Then T-1 is also .

Proof : T is a surjective continuous operator between the Banach spaces X and Y. Therefore, by the open mapping theorem, T takes open sets to open sets.

So, for every open set UX, T(U) is open in Y.

Hence (T-1)-1(U) is open in Y, which proves that T-1 is continuous, i.e. boundedPlanetmathPlanetmathPlanetmathPlanetmath.

0.0.1 Remark

It is usually of great importance to know if a bounded operator T:XY has a bounded inverse. For example, suppose the equation


has unique solutions x for every given yY. Suppose also that the above equation is very difficult to solve (numerically) for a given y0, but easy to solve for a value y~ ”near” y0. Then, if T-1 is continuous, the correspondent solutions x0 and x~ are also ”near” since


Therefore we can solve the equation for a ”near” value y~ instead, without obtaining a significant error.

Title bounded inverse theorem
Canonical name BoundedInverseTheorem
Date of creation 2013-03-22 17:30:51
Last modified on 2013-03-22 17:30:51
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 11
Author asteroid (17536)
Entry type Corollary
Classification msc 47A05
Classification msc 46A30
Synonym inverse mapping theorem