- Bounded inverse theorem
In
mathematics , the bounded inverse theorem is a result in the theory ofbounded linear operator s onBanach space s. 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 theopen mapping theorem and theclosed graph theorem .It is necessary that the spaces in question be Banach spaces. For example, consider the space "X" of
sequence s "x" : N → R with only finitely many non-zero terms equipped with thesupremum norm . The map "T" : "X" → "X" defined by:
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:
converges as "n" → ∞ to the sequence "x"(∞) given by
:
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)
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