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

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 – Map that satisfies a condition similar to that of being an open map.Pages displaying short descriptions of redirect targets
• Closed graph – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets
• 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

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.