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A general theory of linear integral equations of the second kind,

$$ \tag{1 }
\phi ( x) + \int\limits _ { a } ^ { b }  K ( x, s) \phi ( s)  ds  =  f ( x),
$$

established by D. Hilbert [[#References|[1]]] on the basis of his theory of linear and bilinear forms in an infinite number of variables. The principal idea of the theory is as follows. Let there be given a complete orthonormal system of functions  $  \{ \omega _ {n} ( x) \} $
on the interval  $  ( a, b) $
and let

$$ 
\phi _ {p}  =  \int\limits _ { a } ^ { b }  \phi ( t) \omega _ {p} ( t)  dt,\ \ 
f _ {p}  =  \int\limits _ { a } ^ { b }  f ( t) \omega _ {p} ( t)  dt,
$$

$$ 
a _ {pq}  =  \int\limits _ { a } ^ { b }  \int\limits _ { a } ^ { b }  K
( x, t) \omega _ {p} ( t) \omega _ {q} ( t)  dx  dt.
$$

Solving the integral equation (1) is then equivalent to solving the infinite system of linear algebraic equations

$$ \tag{2 }
\phi _ {p} + \sum _ {q = 1 } ^  \infty  
a _ {pq} \phi _ {q}  =  f _ {p} ,\ \ 
p = 1, 2 ,\dots .
$$

Here only those solutions of the system for which

$$ 
\sum _ {p = 1 } ^  \infty   \phi _ {p}  ^ {2}  <  + \infty
$$

are considered, i.e. the system is considered in a [[Hilbert space|Hilbert space]]. The study of (2) in a Hilbert space makes it possible to study the properties of equation (1).

Hilbert's theory of integral equations gives a foundation for the extremal properties of the eigen values of integral equations with a [[Hermitian kernel|Hermitian kernel]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Hilbert,   "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen" , Chelsea, reprint  (1953)</TD></TR></table>

====Comments====
If  $  K $
is Hermitian and the  $  \{ \omega _ {n} \} $
are chosen as a complete orthonormal system of eigen functions corresponding to the eigen values  $  \lambda _ {n} $
of the integral operator, then the system (2) becomes diagonal; solving it leads to the representation

$$ 
\phi  =  \sum _ { p= } 1 ^  \infty   
\frac{f _ {p} }{1+ \lambda _ {p} }

\omega _ {p}  $$

if the conditions of the [[Fredholm alternative|Fredholm alternative]] are fulfilled. If instead of (1) the more general equation

$$ 
\lambda \phi ( x) + \int\limits _ { a } ^ { b }  K ( x, s) \phi ( s)  ds  =  f( x)
$$

is considered, then

$$ 
\phi  =  \sum _ { p= } 1 ^  \infty   
\frac{f _ {p} }{1+ \lambda _ {p} }

\omega _ {p} ,
$$

which holds also for  $  \lambda = 0 $, 
i.e. for the case of integral equations of the first kind (see [[Hilbert–Schmidt series|Hilbert–Schmidt series]]).

The important thing is to consider the concrete integral equation (1) as an abstract linear operator equation on the Hilbert space  $  L _ {2} [ a, b ] $, 
so that the whole theory of Hilbert spaces, of which the discussion in the main article above is just one aspect, is available.

For more information and additional references see [[Integral equation with symmetric kernel|Integral equation with symmetric kernel]].