View source for Fredholm kernel

A Fredholm kernel is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414401.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414402.png" /> giving rise to a [[Completely-continuous operator|completely-continuous operator]]

<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/f/f041/f041440/f0414403.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414404.png" /> is a measurable set in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414405.png" />-dimensional Euclidean space, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414407.png" /> are function spaces. The operator (*) is called a Fredholm integral operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414408.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f0414409.png" />. An important class of Fredholm kernels is that of the measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144011.png" /> for which

<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/f/f041/f041440/f04144012.png" /></td> </tr></table>

A Fredholm kernel that satisfies this condition is also called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144014.png" />-kernel.

A Fredholm kernel is called degenerate if it can be represented as the sum of a product of functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144015.png" /> alone by functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144016.png" /> alone:

<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/f/f041/f041440/f04144017.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144018.png" /> for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144019.png" />, then the Fredholm kernel is called symmetric, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144020.png" />, it is called Hermitian (here the bar denotes complex conjugation). A Fredholm kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144021.png" /> is called skew-Hermitian if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144022.png" />.

The Fredholm kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144024.png" /> are called transposed or allied, and the kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144026.png" /> are called adjoint.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Smirnov,   "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  pp. Chapt. 1  (Translated from Russian)</TD></TR></table>



====Comments====
A completely-continuous operator is nowadays usually called a [[Compact operator|compact operator]].

In the main article above, no distinction is made between real-valued and complex-valued kernels. Usually, symmetry is defined for real-valued kernels, as is skew-symmetry: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144027.png" />. Hermiticity and skew-Hermiticity are then properties of complex-valued kernels. However, the terminology in the literature varies wildly.

About the terminology allied (transposed) and adjoint see also (the editorial comments to) [[Fredholm theorems|Fredholm theorems]].

A Fredholm kernel is a bivalent tensor (cf. [[Tensor on a vector space|Tensor on a vector space]]) giving rise to a [[Fredholm-operator(2)|Fredholm operator]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144029.png" /> be locally convex spaces (cf. [[Locally convex space|Locally convex space]]), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144030.png" /> be the completion of the [[Tensor product|tensor product]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144031.png" /> of these spaces in the inductive topology, that is, in the strongest locally convex topology in which the canonical bilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144032.png" /> is continuous. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144033.png" /> is called a Fredholm kernel if it can be represented in the form

<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/f/f041/f041440/f04144034.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144035.png" /> is a summable sequence of numbers, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144037.png" /> are sequences of elements in some complete convex circled bounded sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144039.png" />, respectively. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144040.png" /> is the dual (cf. [[Adjoint space|Adjoint space]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144041.png" /> of a locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144042.png" />. Then a Fredholm kernel gives rise to a Fredholm operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144043.png" /> of the form

<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/f/f041/f041440/f04144044.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144045.png" /> is the value of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144046.png" /> at the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144047.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144049.png" /> are Banach spaces, then every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144050.png" /> is a Fredholm kernel.

The concept of a Fredholm kernel can also be generalized to the case of the tensor product of several locally convex spaces. Fredholm kernels and Fredholm operators constitute a natural domain of application of the Fredholm theory.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,   "La théorie de Fredholm"  ''Bull. Amer. Math. Soc.'' , '''84'''  (1956)  pp. 319–384</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck,   "Produits tensoriels topologiques et espaces nucleaires"  ''Mem. Amer. Math. Soc.'' , '''5'''  (1955)</TD></TR></table>

''G.L. Litvinov''

====Comments====
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144051.png" /> in a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144052.png" /> over a normal field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144053.png" /> is called circled (or balanced) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144054.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144055.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041440/f04144056.png" />.