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A Fredholm kernel is a function  $  K ( x, y) $
defined on  $  \Omega \times \Omega $
giving rise to a [[Completely-continuous operator|completely-continuous operator]]

$$ \tag{* }
K \phi  \equiv \ 
\int\limits _  \Omega 
K ( x, y) \phi ( y) \ 
dy:  E  \rightarrow  E _ {1} ,
$$

where  $  \Omega $
is a measurable set in an  $  n $-
dimensional Euclidean space, and  $  E $
and  $  E _ {1} $
are function spaces. The operator (*) is called a Fredholm integral operator from  $  E $
into  $  E _ {1} $. 
An important class of Fredholm kernels is that of the measurable functions  $  K ( x, y) $
on  $  \Omega \times \Omega $
for which

$$ 
\int\limits _  \Omega 
\int\limits _  \Omega 
| K ( x, y) |  ^ {2} \ 
dx  dy  <  + \infty .
$$

A Fredholm kernel that satisfies this condition is also called an  $  L _ {2} $-
kernel.

A Fredholm kernel is called degenerate if it can be represented as the sum of a product of functions of  $  x $
alone by functions of  $  y $
alone:

$$ 
K ( x, y)  = \ 
\sum _ {k = 1 } ^ { m } 
\alpha _ {k} ( x)
\beta _ {k} ( y).
$$

If  $  K ( x, y) = K ( y, x) $
for almost-all  $  ( x, y) \in \Omega \times \Omega $, 
then the Fredholm kernel is called symmetric, and if  $  K ( x , y ) = \overline{ {K ( y, x) }}\; $, 
it is called Hermitian (here the bar denotes complex conjugation). A Fredholm kernel  $  K ( x, y) $
is called skew-Hermitian if  $  \overline{ {K ( x, y) }}\; = - K ( y, x) $.

The Fredholm kernels  $  K ( x, y) $
and  $  K ( y, x) $
are called transposed or allied, and the kernels  $  K ( x, y) $
and  $  \overline{ {K ( y, x) }}\; $
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:  $  K ( x , y ) = - K ( y , x ) $. 
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  $  E $
and  $  F $
be locally convex spaces (cf. [[Locally convex space|Locally convex space]]), and let  $  E \overline \otimes \; F $
be the completion of the [[Tensor product|tensor product]]  $  E \otimes F $
of these spaces in the inductive topology, that is, in the strongest locally convex topology in which the canonical bilinear mapping  $  E \times F \rightarrow E \overline \otimes \; F $
is continuous. An element  $  u \in E \overline \otimes \; F $
is called a Fredholm kernel if it can be represented in the form

$$ 
u  =  \sum _ {i = 1 } ^  \infty  
\lambda _ {i} e _ {i} \otimes f _ {i} ,
$$

where  $  \{ \lambda _ {i} \} $
is a summable sequence of numbers, and  $  \{ e _ {i} \} $
and  $  \{ f _ {i} \} $
are sequences of elements in some complete convex circled bounded sets in  $  E $
and  $  F $, 
respectively. Suppose that  $  E $
is the dual (cf. [[Adjoint space|Adjoint space]])  $  G  ^  \prime  $
of a locally convex space  $  G $. 
Then a Fredholm kernel gives rise to a Fredholm operator  $  A:  G \rightarrow F $
of the form

$$ 
x  \rightarrow \ 
\sum _ {i = 1 } ^  \infty  
\lambda _ {i} \langle  x, e _ {i} \rangle f _ {i} ,
$$

where  $  \langle  x, e _ {i} \rangle $
is the value of the functional  $  e _ {i} \in G  ^  \prime  $
at the element  $  x \in G $. 
If  $  E $
and  $  F $
are Banach spaces, then every element of  $  E \overline \otimes \; F $
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 {{ZBL|0073.10101}}</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires"  ''Mem. Amer. Math. Soc.'' , '''5'''  (1955)</TD></TR>
</table>

''G.L. Litvinov''

====Comments====
A set  $  A $
in a topological vector space  $  E $
over a normal field  $  K $
is called circled (or balanced) if  $  k A \subset  A $
for all  $  | k | \leq  1 $
in  $  K $.