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.

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 ℓ_p 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
• Open mapping theorem (functional analysis)
• Surjection of Fréchet spaces
• Webbed space

References

1. Narici 2011, p. 469. sfn error: no target: CITEREFNarici2011 (help)

• Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.CS1 maint: ref=harv (link)
• 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.CS1 maint: ref=harv (link)