Bounded inverse theorem

Bounded inverse theorem In mathematics, the bounded inverse theorem (or inverse mapping 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.

Contents 1 Generalization 2 Counterexample 3 See also 4 References 5 Bibliography Generalization Theorem[1] — If A : X → Y is a continuous linear bijection from a complete pseudometrizable topological vector space (TVS) onto a Hausdorff TVS that is a Baire space, then A : X → Y is a homeomorphism (and thus an isomorphism of TVSs).

Counterexample This theorem may not hold for normed spaces that are not complete. 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 {displaystyle Tx=left(x_{1},{frac {x_{2}}{2}},{frac {x_{3}}{3}},dots right)} is bounded, linear and invertible, but T−1 is unbounded. This does not contradict the bounded inverse theorem since X is not complete, and thus is not a Banach space. To see that it's not complete, consider the sequence of sequences x(n) ∈ X given by {displaystyle x^{(n)}=left(1,{frac {1}{2}},dots ,{frac {1}{n}},0,0,dots right)} converges as n → ∞ to the sequence x(∞) given by {displaystyle x^{(infty )}=left(1,{frac {1}{2}},dots ,{frac {1}{n}},dots right),} which has all its terms non-zero, and so does not lie in X.

The completion of X is the space {displaystyle c_{0}} of all sequences that converge to zero, which is a (closed) subspace of the ℓp space ℓ∞(N), which is the space of all bounded sequences. However, in this case, the map T is not onto, and thus not a bijection. To see this, one need simply note that the sequence {displaystyle x=left(1,{frac {1}{2}},{frac {1}{3}},dots right),} is an element of {displaystyle c_{0}} , but is not in the range of {displaystyle T:c_{0}to c_{0}} .

See also Almost open linear map Closed graph Closed graph theorem – Theorem relating continuity to graphs Open mapping theorem (functional analysis) – Condition for a linear operator to be open Surjection of Fréchet spaces – Characterization of surjectivity Webbed space – Space where open mapping and closed graph theorems hold References ^ Narici & Beckenstein 2011, p. 469. Bibliography Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. 356. ISBN 0-387-00444-0. (Section 8.2) Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. show vte Functional analysis (topics – glossary) show vte Topological vector spaces (TVSs) Categories: Operator theoryTheorems in functional analysis

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