In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the bounded inverse theorem (or inverse mapping theorem) is a result in the theory of
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
s on
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s.
It states that a
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
bounded linear operator ''T'' from one Banach space to another has bounded
inverse ''T''
−1. It is
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
to both the
open mapping theorem and the
closed graph theorem
In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs.
Each gives conditions when functions with closed graphs are necessarily continuous.
Graphs and m ...
.
Generalization
Counterexample
This theorem may not hold for normed spaces that are not complete.
For example, consider the space ''X'' of
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s ''x'' : N → R with only finitely many non-zero terms equipped with the
supremum norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when the ...
. 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
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
, 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
.
See also
*
*
*
*
*
*
References
Bibliography
*
*
* (Section 8.2)
*
{{Topological vector spaces
Operator theory
Theorems in functional analysis