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 |
Orthonormal system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthonormal_system&oldid=41849