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A statement of an alternative that follows from the [[Fredholm theorems|Fredholm theorems]]. In the case of a linear Fredholm integral equation of the second kind,

$$ \tag{1 }
\phi ( x) - \lambda
\int\limits _ { a } ^ { b }  K ( x, s) \phi ( s)  ds  =  f ( x),\ \ 
x \in [ a, b],
$$

the Fredholm alternative states that either equation (1) and its conjugate equation

$$ \tag{2 }
\psi ( x) - \overline \lambda \;
\int\limits _ { a } ^ { b }  \overline{ {K ( s, x) }}\; \psi ( s)  ds  =  g ( x),\ \ 
x \in [ a, b],
$$

have unique solutions  $  \phi , \psi $, 
for any given functions  $  f $
and  $  g $, 
or the corresponding homogeneous equations

$$ \tag{1'}
\phi ( x) - \lambda
\int\limits _ { a } ^ { b }  K ( x, s) \phi ( s)  ds  =  0,
$$

$$ \tag{2'}
\psi ( x) - \overline \lambda \; \int\limits _ { a } ^ { b }  \overline{ {K ( s, x) }}\; \psi ( s)  ds  =  0
$$

have non-zero solutions, where the number of linearly independent solutions is finite and is the same for both equations.

In the second case equation (1) has a solution if and only if

$$ 
\int\limits _ { a } ^ { b }  f ( x) \overline{ {\psi _ {k} ( x) }}\;  dx  =  0,\ \ 
k = 1 \dots n,
$$

where  $  \psi _ {1} \dots \psi _ {n} $
is a complete system of linearly independent solutions of (2'}). Here the general solution of (1) has the form

$$ 
\phi ( x)  = \ 
\phi _ {*} ( x) +
\sum _ {k = 1 } ^ { n }  c _ {k} \phi _ {k} ( x),
$$

where  $  \phi _ {*} $
is some solution of (1),  $  \phi _ {1} \dots \phi _ {n} $
is a complete system of linearly independent solutions of (1'}), and the  $  c _ {k} $
are arbitrary constants. Similar statements also hold for equation (2).

Let  $  T $
be a continuous linear operator mapping a Banach space  $  E $
into itself; let  $  E  ^ {*} $
and  $  T  ^ {*} $
be the corresponding dual space and dual operator. Consider the equations:

$$ \tag{3 }
T ( x)  =  y,\ \ 
x, y \in E,
$$

$$ \tag{4 }
T  ^ {*} ( g)  =  f,\  g, f \in E  ^ {*} ,
$$

$$ \tag{3'}
T ( x)  =  0,\  x \in E,
$$

$$ \tag{4'}
T  ^ {*} ( g)  =  0,\  g \in E  ^ {*} .
$$

The Fredholm alternative for  $  T $
means the following: 1) either the equations (3) and (4) have solutions, for arbitrary right-hand sides, and then their solutions are unique; or 2) the homogeneous equations (3'}) and (4'}) have the same finite number of linearly independent solutions  $  x _ {1} \dots x _ {n} $
and  $  g _ {1} \dots g _ {n} $, 
respectively; in this case, for equation (3), or (4) respectively, to have a solution, it is necessary and sufficient that  $  g _ {k} ( y) = 0 $, 
$  k = 1 \dots n $, 
or  $  f ( x _ {k} ) = 0 $, 
$  k = 1 \dots n $, 
respectively; here the general solution of (3) is given by

$$ 
x  =  x  ^ {*} +
\sum _ {k = 1 } ^ { n }  c _ {k} x _ {k} ,
$$

and the general solution of (4) by

$$ 
g  =  g  ^ {*} +
\sum _ {k = 1 } ^ { n }  c _ {k} g _ {k} ,
$$

where  $  x  ^ {*} $(
respectively,  $  g  ^ {*} $) 
is some solution of (3) ((4)), and  $  c _ {1} \dots c _ {n} $
are arbitrary constants.

Each of the following two conditions is necessary and sufficient for the Fredholm alternative to hold for the operator  $  T $.

1)  $  T $
can be represented in the form

$$ 
T  =  W + V,
$$

where  $  W $
is an operator with a two-sided continuous inverse and  $  V $
is a compact operator.

2)  $  T $
can be represented in the form

$$ 
T  =  W _ {1} + V _ {1} ,
$$

where  $  W _ {1} $
is an operator with a two-sided continuous inverse and  $  V _ {1} $
is a finite-dimensional operator.

====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><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,   "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Kantorovich,   G.P. Akilov,   "Functional analysis in normed spaces" , Pergamon  (1964)  (Translated from Russian)</TD></TR></table>

====Comments====
The precise form of the Fredholm alternative is as follows: Consider the equations (1) and (1'}) with a continuous kernel  $  K $. 
Then either equation (1) has a continuous solution  $  \phi $
for any right-hand side  $  f $
or the homogeneous equation (1'}) has a non-trivial solution. In abstract form the alternative may be stated as follows. For a [[Fredholm-operator(2)|Fredholm operator]]  $  T $
of index zero (cf. [[Index of an operator|Index of an operator]]) acting on a [[Banach space|Banach space]] the following holds true: Either  $  T $
is invertible or  $  T $
has a non-trivial kernel (cf. [[Kernel of a linear operator|Kernel of a linear operator]]; [[Kernel of an integral operator|Kernel of an integral operator]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. Gohberg,   S. Goldberg,   "Basic operator theory" , Birkhäuser  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.E. Taylor,   D.C. Lay,   "Introduction to functional analysis" , Wiley  (1980)  pp. Chapt. 5</TD></TR></table>