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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 ''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 : 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 : converges as ''n'' → ∞ to the sequence ''x''(∞) given by : which has all its terms non-zero, and so does not lie in ''X''. The completion of ''X'' is the space 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 : is an element of , but is not in the range of . ==References== * (Section 8.2) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「bounded inverse theorem」の詳細全文を読む スポンサード リンク
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