Bounded inverse theorem

In mathematics, the bounded inverse theorem ( also called inverse mapping theorem or Banach isomorphism 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 — 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

[math]\displaystyle{ T x = \left( x_{1}, \frac{x_{2}}{2}, \frac{x_{3}}{3}, \dots \right) }[/math]

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

[math]\displaystyle{ x^{(n)} = \left( 1, \frac1{2}, \dots, \frac1{n}, 0, 0, \dots \right) }[/math]

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

[math]\displaystyle{ x^{(\infty)} = \left( 1, \frac1{2}, \dots, \frac1{n}, \dots \right), }[/math]

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

The completion of X is the space [math]\displaystyle{ c_0 }[/math] 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

[math]\displaystyle{ x = \left( 1, \frac12, \frac13, \dots \right), }[/math]

is an element of [math]\displaystyle{ c_0 }[/math], but is not in the range of [math]\displaystyle{ T:c_0\to c_0 }[/math].

See also

• Almost open linear map
• Closed graph – Graph of a map closed in the product space
• 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

Bibliography

• Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. 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. https://archive.org/details/introductiontopa00roge_558.  (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.