Anderson orthogonality theorem

In mathematics, the bounded inverse 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.

It is necessary that the spaces in question be Banach spaces. 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

$Tx=\left(x_{1},{\frac {x_{2}}{2}},{\frac {x_{3}}{3}},\dots \right)$

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

$x^{(n)}=\left(1,{\frac {1}{2}},\dots ,{\frac {1}{n}},0,0,\dots \right)$

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

$x^{(\infty )}=\left(1,{\frac {1}{2}},\dots ,{\frac {1}{n}},\dots \right),$

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

The completion of X is the space $c_{0}$ 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

$x=\left(1,{\frac {1}{2}},{\frac {1}{3}},\dots \right),$

is an element of $c_{0}$, but is not in the range of $T:c_{0}\to c_{0}$.

References

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