# Bounded inverse theorem

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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 T1. It is equivalent to both the open mapping theorem and the closed graph theorem.

## Generalization

Theorem [1]   If A : XY is a continuous linear bijection from a complete Pseudometrizable topological vector space (TVS) onto a Hausdorff TVS that is a Baire space, then A : XY 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 T1 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 p space(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 ${\displaystyle T:c_{0}\to c_{0}}$.

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