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''(inverse operator) of a single-valued onto mapping (operator)
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''(inverse operator) of a single-valued onto mapping (operator) $ f: M \to f[M] $''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i0523801.png" /></td> </tr></table>
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A single-valued mapping $ g $ such that
 +
\begin{align}
 +
g \circ f & = \mathsf{Id}_{X} \quad \text{on} ~ M, \qquad (1) \\
 +
f \circ g & = \mathsf{Id}_{Y} \quad \text{on} ~ f[M], \qquad (2)
 +
\end{align}
 +
where $ M \subseteq X $, $ f[M] \subseteq Y $, and $ X $ and $ Y $ are any sets.
  
''
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If $ g $ satisfies only Condition (1), then it is called a '''left-inverse''' mapping of $ f $, and if it satisfies only Condition (2), then it is called a '''right-inverse''' mapping of $ f $. The inverse mapping $ f^{-1} $ exists if and only if for each $ y \in f[M] $, the inverse image $ {f^{\leftarrow}}[\{ y \}] $ consists of just a single element $ x \in M $. If $ f $ has an inverse mapping $ f^{-1} $, then the equation
 +
$$
 +
f(x) = y \qquad (3)
 +
$$
 +
has a unique solution for each $ y \in f[M] $. If only a right inverse $ f_{R}^{-1} $ exists, then a solution of (3) exists, but its uniqueness is an open question. If only a left inverse $ f_{L}^{-1} $ exists, then any solution is unique, assuming that it exists.
  
A single-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i0523802.png" /> such that
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If $ X $ and $ Y $ are vector spaces, and if $ A $ is a linear operator from $ X $ into $ Y $, then $ A^{-1} $ is also linear, if it exists. In general, if $ X $ and $ Y $ are endowed with some kind of structure, it may happen that certain properties of $ A $ are also inherited by $ A^{-1} $, assuming that it exists. For example:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i0523803.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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* If $ X $ and $ Y $ are Banach spaces, and if $ A: X \to Y $ is a closed operator, then $ A^{-1} $ is also closed.
 +
* If $ \mathcal{H} $ is a Hilbert space and $ A: \mathcal{H} \to \mathcal{H} $ is self-adjoint, then $ A^{-1} $ is also self-adjoint.
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* If $ f: \mathbf{R} \to \mathbf{R} $ is an odd function, then $ f^{-1} $ is also odd.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i0523804.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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The continuity of $ A $ does not always imply the continuity of $ A^{-1} $ for many important classes of linear operators, e.g., completely-continuous operators. The following are important tests for the continuity of the inverse of a linear operator.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i0523805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i0523806.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i0523807.png" /> are any sets.
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Let $ X $ be a finite-dimensional vector space, with a certain basis, and let $ A: X \to X $ be given by the matrix $ [a_{ij}] $ with respect to this basis. Then $ A^{-1} $ exists if and only if $ \det([a_{ij}]) \neq 0 $ (in this case, $ A $ and $ A^{-1} $ are automatically continuous).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i0523808.png" /> satisfies only condition (1), then it is called a left-inverse mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i0523809.png" />, and if it satisfies only condition (2), it is a right-inverse mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238010.png" />. The inverse mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238011.png" /> exists if and only if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238012.png" /> the complete inverse image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238013.png" /> consists of a single element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238015.png" /> has an inverse mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238016.png" />, then the equation
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Let $ X $ and $ Y $ be Banach spaces, and let $ A $ be a continuous linear operator from $ X $ into $ Y $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
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# If $ \| A(x) \| \geq m \| x \| $, where $ m > 0 $, then $ A^{-1} $ exists and is continuous.
 +
# If $ X = Y $ and $ \| A \| < 1 $, then $ (\mathsf{Id} - A)^{-1} $ exists, is continuous, and $$ (\mathsf{Id} - A)^{-1} = \sum_{n = 0}^{\infty} A^{n}, $$ where the series on the right-hand side converges in the norm of the space $ \mathcal{L}(X) $.
 +
# The operator $ A^{-1} $ exists and is continuous on all of $ Y $ if and only if the conjugate $ A^{*} $ has an inverse that is defined and continuous on $ X^{*} $. Here, $ (A^{-1})^{*} = (A^{*})^{-1} $.
 +
# If $ A^{-1} $ exists and is continuous, and if $ \| A - B \| < \| A^{-1} \|^{-1} $, then $ B^{-1} $ also exists, is continuous, and $$ B^{-1} = A^{-1} \sum_{n = 0}^{\infty} [(A - B) A^{-1}]^{n}. $$ Therefore, the set of invertible operators is open in $ \mathcal{L}(X,Y) $ in the [[Uniform topology|uniform topology]] of this space.
 +
# '''Banach’s Open-Mapping Theorem.''' If $ A $ is a one-to-one mapping of $ X $ onto $ Y $, then the inverse mapping, which exists, is continuous. This theorem has the following generalization: A one-to-one continuous linear mapping of a fully-complete space $ X $ onto a separated barrelled space $ Y $ is a topological [[Isomorphism|isomorphism]].
  
has a unique solution for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238018.png" />. If only a right inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238019.png" /> exists, then a solution of (3) exists, but its uniqueness is an open question. If only a left inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238020.png" /> exists, then the solution is unique, assuming that it exists. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238022.png" /> are vector spaces and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238023.png" /> is a linear operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238024.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238025.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238026.png" /> is also linear, if it exists. In general, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238028.png" /> are endowed with some kind of structure, it may happen that certain properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238029.png" /> are also inherited by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238030.png" />, assuming that it exists. E.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238032.png" /> are Banach spaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238033.png" /> is a closed operator, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238034.png" /> is also closed; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238035.png" /> is a Hilbert space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238036.png" /> is self-adjoint, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238037.png" /> is also self-adjoint; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238038.png" /> is an odd function, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238039.png" /> is also odd, etc. The continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238040.png" /> does not always imply the continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238041.png" /> for many important classes of linear operators, for example for completely-continuous operators. The following are important tests for the continuity of the inverse of a linear operator.
+
The spectral theory of linear operators on a Hilbert space contains a number of results on the existence and continuity of the inverse of a continuous linear operator. For example, if $ A $ is self-adjoint and $ \lambda $ is not real, then $ (A - \lambda \cdot \mathsf{Id})^{-1} $ exists and is continuous.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238042.png" /> be a finite-dimensional vector space, with a certain basis, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238043.png" /> be given by the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238044.png" /> with respect to this basis. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238045.png" /> exists if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238046.png" /> (in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238048.png" /> are automatically continuous).
+
====References====
 
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238050.png" /> be Banach spaces, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238051.png" /> be a continuous linear operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238052.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238053.png" />.
 
 
 
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238055.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238056.png" /> exists and is continuous.
 
  
2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238058.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238059.png" /> exists, is continuous and
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<table>
 
+
<TR><TD valign="top">[1]</TD><TD valign="top">
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238060.png" /></td> </tr></table>
+
N. Dunford, J.T. Schwartz, “Linear operators. General theory”, '''1''', Interscience (1958).</TD></TR>
 
+
<TR><TD valign="top">[2]</TD> <TD valign="top">
where the series on the right-hand side converges in the norm of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238061.png" />.
+
L.V. Kantorovich, G.P. Akilov, “Functional analysis”, Pergamon (1982). (Translated from Russian)</TD></TR>
 
+
<TR><TD valign="top">[3]</TD><TD valign="top">
3) The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238062.png" /> exists and is continuous on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238063.png" /> if and only if the conjugate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238064.png" /> has an inverse which is defined and continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238065.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238066.png" />.
+
W. Rudin, “Functional analysis”, McGraw-Hill (1979).</TD></TR>
 
+
<TR><TD valign="top">[4]</TD><TD valign="top">
4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238067.png" /> exists, is continuous and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238068.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238069.png" /> also exists, is continuous and
+
A.P. Robertson, W.S. Robertson, “Topological vector spaces”, Cambridge Univ. Press (1964).</TD></TR>
 
+
</table>
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238070.png" /></td> </tr></table>
 
 
 
Thus, the set of invertible operators is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238071.png" /> in the [[Uniform topology|uniform topology]] of this space.
 
 
 
5) Banach's open mapping theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238072.png" /> is a one-to-one mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238073.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238074.png" />, then the inverse mapping, which exists, is continuous. This theorem has the following generalization: A one-to-one continuous linear mapping of a fully-complete space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238075.png" /> onto a separated barrelled space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238076.png" /> is a topological [[Isomorphism|isomorphism]].
 
 
 
The spectral theory of linear operators on a Hilbert space contains a number of results on the existence and continuity of the inverse of a continuous linear operator. E.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238077.png" /> is self-adjoint and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238078.png" /> is not real, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052380/i05238079.png" /> exists and is continuous.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Dunford,   J.T. Schwartz,   "Linear operators. General theory" , '''1''' , Interscience (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.V. Kantorovich,   G.P. Akilov,   "Functional analysis" , Pergamon (1982) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W. Rudin,   "Functional analysis" , McGraw-Hill (1979)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.P. Robertson,   W.S. Robertson,   "Topological vector spaces" , Cambridge Univ. Press (1964)</TD></TR></table>
 

Latest revision as of 03:29, 9 January 2017

(inverse operator) of a single-valued onto mapping (operator) $ f: M \to f[M] $

A single-valued mapping $ g $ such that \begin{align} g \circ f & = \mathsf{Id}_{X} \quad \text{on} ~ M, \qquad (1) \\ f \circ g & = \mathsf{Id}_{Y} \quad \text{on} ~ f[M], \qquad (2) \end{align} where $ M \subseteq X $, $ f[M] \subseteq Y $, and $ X $ and $ Y $ are any sets.

If $ g $ satisfies only Condition (1), then it is called a left-inverse mapping of $ f $, and if it satisfies only Condition (2), then it is called a right-inverse mapping of $ f $. The inverse mapping $ f^{-1} $ exists if and only if for each $ y \in f[M] $, the inverse image $ {f^{\leftarrow}}[\{ y \}] $ consists of just a single element $ x \in M $. If $ f $ has an inverse mapping $ f^{-1} $, then the equation $$ f(x) = y \qquad (3) $$ has a unique solution for each $ y \in f[M] $. If only a right inverse $ f_{R}^{-1} $ exists, then a solution of (3) exists, but its uniqueness is an open question. If only a left inverse $ f_{L}^{-1} $ exists, then any solution is unique, assuming that it exists.

If $ X $ and $ Y $ are vector spaces, and if $ A $ is a linear operator from $ X $ into $ Y $, then $ A^{-1} $ is also linear, if it exists. In general, if $ X $ and $ Y $ are endowed with some kind of structure, it may happen that certain properties of $ A $ are also inherited by $ A^{-1} $, assuming that it exists. For example:

  • If $ X $ and $ Y $ are Banach spaces, and if $ A: X \to Y $ is a closed operator, then $ A^{-1} $ is also closed.
  • If $ \mathcal{H} $ is a Hilbert space and $ A: \mathcal{H} \to \mathcal{H} $ is self-adjoint, then $ A^{-1} $ is also self-adjoint.
  • If $ f: \mathbf{R} \to \mathbf{R} $ is an odd function, then $ f^{-1} $ is also odd.

The continuity of $ A $ does not always imply the continuity of $ A^{-1} $ for many important classes of linear operators, e.g., completely-continuous operators. The following are important tests for the continuity of the inverse of a linear operator.

Let $ X $ be a finite-dimensional vector space, with a certain basis, and let $ A: X \to X $ be given by the matrix $ [a_{ij}] $ with respect to this basis. Then $ A^{-1} $ exists if and only if $ \det([a_{ij}]) \neq 0 $ (in this case, $ A $ and $ A^{-1} $ are automatically continuous).

Let $ X $ and $ Y $ be Banach spaces, and let $ A $ be a continuous linear operator from $ X $ into $ Y $.

  1. If $ \| A(x) \| \geq m \| x \| $, where $ m > 0 $, then $ A^{-1} $ exists and is continuous.
  2. If $ X = Y $ and $ \| A \| < 1 $, then $ (\mathsf{Id} - A)^{-1} $ exists, is continuous, and $$ (\mathsf{Id} - A)^{-1} = \sum_{n = 0}^{\infty} A^{n}, $$ where the series on the right-hand side converges in the norm of the space $ \mathcal{L}(X) $.
  3. The operator $ A^{-1} $ exists and is continuous on all of $ Y $ if and only if the conjugate $ A^{*} $ has an inverse that is defined and continuous on $ X^{*} $. Here, $ (A^{-1})^{*} = (A^{*})^{-1} $.
  4. If $ A^{-1} $ exists and is continuous, and if $ \| A - B \| < \| A^{-1} \|^{-1} $, then $ B^{-1} $ also exists, is continuous, and $$ B^{-1} = A^{-1} \sum_{n = 0}^{\infty} [(A - B) A^{-1}]^{n}. $$ Therefore, the set of invertible operators is open in $ \mathcal{L}(X,Y) $ in the uniform topology of this space.
  5. Banach’s Open-Mapping Theorem. If $ A $ is a one-to-one mapping of $ X $ onto $ Y $, then the inverse mapping, which exists, is continuous. This theorem has the following generalization: A one-to-one continuous linear mapping of a fully-complete space $ X $ onto a separated barrelled space $ Y $ is a topological isomorphism.

The spectral theory of linear operators on a Hilbert space contains a number of results on the existence and continuity of the inverse of a continuous linear operator. For example, if $ A $ is self-adjoint and $ \lambda $ is not real, then $ (A - \lambda \cdot \mathsf{Id})^{-1} $ exists and is continuous.

References

[1] N. Dunford, J.T. Schwartz, “Linear operators. General theory”, 1, Interscience (1958).
[2] L.V. Kantorovich, G.P. Akilov, “Functional analysis”, Pergamon (1982). (Translated from Russian)
[3] W. Rudin, “Functional analysis”, McGraw-Hill (1979).
[4] A.P. Robertson, W.S. Robertson, “Topological vector spaces”, Cambridge Univ. Press (1964).
How to Cite This Entry:
Inverse mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inverse_mapping&oldid=15880
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article