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An equality expressing the square of the norm of an element in a vector space with a scalar product in terms of the square of the moduli of the [[Fourier coefficients|Fourier coefficients]] of this element in some [[Orthogonal system|orthogonal system]]. Thus, if  $  X $
is a normed separable vector space with a scalar product  $  (  , ) $, 
if  $  \| \cdot \| $
is the corresponding norm and if  $  \{ e _ {n} \} $
is an orthogonal system in  $  X $, 
$  e _ {n} \neq 0 $, 
$  n = 1, 2 \dots $
then Parseval's equality for an element  $  x \in X $
is

$$ \tag{1 }
\| x \|  ^ {2}  =  \sum_{n=1} ^  \infty   | a _ {n} |  ^ {2} \| e _ {n} \|  ^ {2} ,
$$

where  $  a _ {n} = ( x, e _ {n} )/( e _ {n} , e _ {n} ) $, 
$  n = 1, 2 \dots $
are the Fourier coefficients of  $  x $
in the system  $  \{ e _ {n} \} $. 
If  $  \{ e _ {n} \} $
is orthonormal, then Parseval's equality has the form

$$ 
\| x \|  ^ {2}  =  \sum_{n=1} ^  \infty   | a _ {n} |  ^ {2} .
$$

The validity of Parseval's equality for a given element  $  x \in X $
is a necessary and sufficient condition for its Fourier series in the orthogonal system  $  \{ e _ {n} \} $
to converge to  $  x $
in the norm of  $  X $. 
The validity of Parseval's equality for every element  $  x \in X $
is a necessary and sufficient condition for the orthogonal system  $  \{ e _ {n} \} $
to be complete in  $  X $(
cf. [[Complete system|Complete system]]). This implies, in particular, that:

1) if  $  X $
is a separable Hilbert space (cf. [[Hilbert space|Hilbert space]]) and  $  \{ e _ {n} \} $
is an orthogonal basis of it, then Parseval's equality holds for  $  \{ e _ {n} \} $
for every  $  x \in X $;

2) if  $  X $
is a separable Hilbert space,  $  x , y \in X $, 
if  $  \{ e _ {n} \} $
is an orthonormal basis of  $  X $
and if  $  a _ {n} = ( x, e _ {n} ) $
and  $  b _ {n} = ( y, e _ {n} ) $
are the Fourier coefficients of  $  x $
and  $  y $, 
then

$$ \tag{2 }
( x, y)  =  \sum_{n=1} ^  \infty   a _ {n} \overline{ {b _ {n} }}\; ,
$$

the so-called generalized Parseval equality. In a fairly-definitive form the question of the completeness of a system of functions that are the eigen functions of differential operators was studied by V.A. Steklov in [[#References|[1]]].

Parseval's equality can also be generalized to the case of non-separable Hilbert spaces: If  $  \{ e _  \alpha  \} $, 
$  \alpha \in \mathfrak A $(
$  \mathfrak A $
is a certain index set), is a complete orthonormal system in a Hilbert space  $  X $, 
then for any element  $  x \in X $
Parseval's equality holds:

$$ 
( x, x)  =  \sum _ {\alpha \in \mathfrak A } | ( x, e _  \alpha  ) |  ^ {2} ,
$$

and the sum on the right-hand side is to be understood as

$$ 
\sup _ {\mathfrak A _ {0} }  \sum _ {\alpha \in \mathfrak A } | ( x, e _  \alpha  ) |  ^ {2} ,
$$

where the supremum is taken over all finite subsets  $  \mathfrak A _ {0} $
of  $  \mathfrak A $.

When  $  X = L _ {2} [- \pi , \pi ] $, 
the space of real-valued functions with Lebesgue-integrable squares on  $  [- \pi , \pi ] $, 
and  $  f \in L _ {2} [- \pi , \pi ] $, 
then one may take the [[Trigonometric system|trigonometric system]] as a complete orthogonal system and

$$ 
f  \sim  
\frac{a _ {0} }{2}
 + \sum_{n=1} ^  \infty  
( a _ {n}  \cos  nx + b _ {n}  \sin  nx),
$$

where (1) takes the form

$$ 

\frac{1} \pi 
 \int\limits _ {- \pi } ^  \pi   f ^ { 2 } ( t)  dt  = \ 

\frac{a _ {0}  ^ {2} }{2}
 + \sum_{n=1} ^  \infty   ( a _ {n}  ^ {2} + b _ {n}  ^ {2}
),
$$

which is called the classical Parseval equality. It was proved in 1805 by M. Parseval.

If  $  g \in L _ {2} [- \pi , \pi ] $
and

$$ 
g  \sim  
\frac{a _ {0}  ^  \prime  }{2}
 + \sum_{n=1} ^  \infty  
( a _ {n}  ^  \prime   \cos  nx + b _ {n}  ^  \prime   \sin  nx ),
$$

then an equality similar to (2) looks as follows:

$$ \tag{3 }

\frac{1} \pi 
 \int\limits _ {- \pi } ^  \pi   f( t) g( t)  dt  = \ 

\frac{1}{2}
 a _ {0} a _ {0}  ^  \prime  + \sum_{n=1} ^  \infty   ( a _ {n} a _ {n}  ^  \prime  +
b _ {n} b _ {n}  ^  \prime  ).
$$

Two classes  $  K $
and  $  K  ^  \prime  $
of real-valued functions defined on  $  [- \pi , \pi ] $
and such that for all  $  f \in K $
and  $  g \in K  ^  \prime  $
Parseval's equality (3) holds are called complementary. An example of complementary classes are the spaces  $  L _ {p} [- \pi , \pi ] $
and  $  L _ {q} [- \pi , \pi ] $, 
$  p  ^ {-} 1 + q  ^ {-} 1 = 1 $, 
$  1 < p < + \infty $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Steklov,   "Sur certaines égalités générales communes à plusieurs séries de fonctions souvent employées dans l'analyse"  ''Zap. Nauchn. Fiz.-Mat. Obshch. Ser. 8'' , '''157'''  (1904)  pp. 1–32</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,   "A course of mathematical analysis" , '''2''' , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Il'in,   E.G. Poznyak,   "Fundamentals of mathematical analysis" , '''2''' , MIR  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.A. Kirillov,   A.D. Gvishiani,   "Theorems and problems in functional analysis" , Springer  (1982)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,   K.R. Stromberg,   "Real and abstract analysis" , Springer  (1965)</TD></TR></table>