Orthonormal system

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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: 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 $$ \sum_{k=1}^\infty |c_k|^2 < \infty \ . $$


[1] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Pitman (1981) (Translated from Russian)
[2] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)

A.A. Talalyan



[a1] J. Weidmann, "Linear operators in Hilbert space" , Springer (1980)
[a2] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5
How to Cite This Entry:
Orthonormal system. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii, A.A. Talalyan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article