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[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
- [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 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
- [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
References
- ↑ Narici & Beckenstein 2011, p. 469.
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.
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