Orthonormal system

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An orthonormal system of vectors is a set of vectors in a Euclidean (Hilbert) space with inner product such that if (orthogonality) and (normalization).

M.I. Voitsekhovskii

An orthonormal system of functions is a system of functions in a space which is simultaneously orthogonal and normalized in , i.e.

(see Normalized system; Orthogonal system). 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

in terms of the behaviour of the coefficients . An example of this type of theorem is the Riesz–Fischer theorem: The series

with respect to an orthonormal system in converges in the metric of if and only if


[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