# Orthonormal system

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

#### References

[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*

#### Comments

#### References

[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: http://encyclopediaofmath.org/index.php?title=Orthonormal_system&oldid=15467