Bounded inverse theorem

In mathematics, the bounded inverse theorem is a result in the theory of bounded linear operators on Banach spaces. It states that a bijective bounded linear operator "T" from one Banach space to another has bounded inverse "T"⁻¹. It is equivalent to both the open mapping theorem and the closed graph theorem.

It is necessary that the spaces in question be Banach spaces. For example, consider the space "X" of sequences "x" : N → R with only finitely many non-zero terms equipped with the supremum norm. The map "T" : "X" → "X" defined by

:$T\; x\; =\; left(\; x\_\{1\},\; frac\{x\_\{2\{2\},\; frac\{x\_\{3\{3\},\; dots\; ight)$

is bounded, linear and invertible, but "T"⁻¹ is unbounded. This does not contradict the bounded inverse theorem since "X" is not a closed linear subspace of the ℓ^"p" space ℓ^∞(N), and hence is not a Banach space. For example, the sequence of sequences "x"^("n") ∈ "X" given by

:$x^\{(n)\}\; =\; left(\; 1,\; frac1\{2\},\; dots,\; frac1\{n\},\; 0,\; 0,\; dots\; ight)$

converges as "n" → ∞ to the sequence "x"^(∞) given by

:$x^\{(infty)\}\; =\; left(\; 1,\; frac1\{2\},\; dots,\; frac1\{n\},\; dots\; ight),$

which has all its terms non-zero, and so does not lie in "X".

References

* cite book
author = Renardy, Michael and Rogers, Robert C.
title = An introduction to partial differential equations
series = Texts in Applied Mathematics 13
edition = Second edition
publisher = Springer-Verlag
location = New York
year = 2004
pages = 356
isbn = 0-387-00444-0 (Section 7.2)