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An orthonormal system of vectors is a system $(x_\alpha)$ of vectors in a Euclidean (Hilbert) space with inner product $(\cdot,\cdot)$ such that $(x_\alpha,x_\beta) = 0$ if $\alpha \ne \beta$ (orthogonality) and $(x_\alpha,x_\alpha) = 1$ (normalization).

''M.I. Voitsekhovskii''

An orthonormal system of functions in a space $L^2(X,S,\mu)$ is a system $(\phi_k)$ of functions which is both an [[orthogonal system]] and a [[normalized system]] , i.e.
$$
\int_X \phi_i \bar\phi_j \mathrm{d}\mu = \begin{cases} 0 & \ \text{if}\ i \ne j \\ 1 & \ \text{if}\ i=j \end{cases} \ .
$$

In the mathematical literature, the term  "orthogonal system"  often means  "orthonormal system" ; when studying a given orthogonal system, it is not always crucial whether or not it is normalized. None the less, if the systems are normalized, a clearer formulation is possible for certain theorems on the convergence of a series
$$
\sum_{k=1}^\infty c_k \phi_k(x)
$$
in terms of the behaviour of the coefficients $(c_k)$. An example of this type of theorem is the [[Riesz-Fischer theorem|Riesz–Fischer theorem]]: The series
$$
\sum_{k=1}^\infty c_k \phi_k(x)
$$
with respect to an orthonormal system$(\phi_k)_{k=1}^\infty$ in $L^2[a,b]$ converges in the metric of $L^2[a,b]$ if and only if
$$
\sum_{k=1}^\infty |c_k|^2 < \infty \ .
$$

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,   "Elements of the theory of functions and functional analysis" , '''1–2''' , Pitman  (1981)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  S. Kaczmarz,   H. Steinhaus,   "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR>
</table>

''A.A. Talalyan''

====Comments====


====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Weidmann,   "Linear operators in Hilbert space" , Springer  (1980)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Yosida,   "Functional analysis" , Springer  (1980)  pp. Chapt. 8, Sect. 4; 5</TD></TR>
</table>

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