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In [[mathematics]], the '''bounded inverse theorem''' is a result in the theory of [[bounded linear operator]]s on [[Banach space]]s. It states that a [[bijection|bijective]] bounded linear operator ''T'' from one Banach space to another has bounded [[inverse function|inverse]] ''T''<sup>&minus;1</sup>. It is [[logical equivalence|equivalent]] to both the [[Open mapping theorem (functional analysis)|open mapping theorem]] and the [[closed graph theorem]].
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It is necessary that the spaces in question be Banach spaces. For example, consider the space ''X'' of [[sequence]]s ''x''&nbsp;:&nbsp;'''N'''&nbsp;&rarr;&nbsp;'''R''' with only finitely many non-zero terms equipped with the [[supremum norm]]. The map ''T''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''X'' defined by
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:<math>T x = \left( x_{1}, \frac{x_{2}}{2}, \frac{x_{3}}{3}, \dots \right)</math>
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is bounded, linear and invertible, but ''T''<sup>&minus;1</sup> is unbounded. This does not contradict the bounded inverse theorem since ''X'' is not [[completeness (topology)|complete]], and thus is not a Banach space. To see that it's not complete, consider the sequence of sequences ''x''<sup>(''n'')</sup>&nbsp;&isin;&nbsp;''X'' given by
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:<math>x^{(n)} = \left( 1, \frac1{2}, \dots, \frac1{n}, 0, 0, \dots \right)</math>
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converges as ''n''&nbsp;&rarr;&nbsp;&infin; to the sequence ''x''<sup>(&infin;)</sup> given by
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:<math>x^{(\infty)} = \left( 1, \frac1{2}, \dots, \frac1{n}, \dots \right),</math>
</ul>
 
which has all its terms non-zero, and so does not lie in ''X''.
 
The completion of ''X'' is the space <math>c_0</math> of all sequences that converge to zero, which is a (closed) subspace of the [[Lp space|ℓ<sub>''p''</sub> space]] ℓ<sub>&infin;</sub>('''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>x = \left( 1, \frac12, \frac13, \dots \right),</math>
 
is an element of <math>c_0</math>, but is not in the range of <math>T:c_0\to c_0</math>.
 
==References==
 
* {{cite book
|   author = Renardy, Michael and Rogers, Robert C.
|   title = An introduction to partial differential equations
|   series = Texts in Applied Mathematics 13
|  edition = Second edition
|publisher = Springer-Verlag
| location = New York
|    year = 2004
|    pages = 356
|    isbn = 0-387-00444-0
}} (Section 8.2)
 
[[Category:Operator theory]]
[[Category:Theorems in functional analysis]]

Latest revision as of 12:45, 12 November 2014

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