Orthogonal basis

A system of pairwise orthogonal non-zero elements of a Hilbert space , such that any element can be (uniquely) represented in the form of a norm-convergent series

called the Fourier series of the element with respect to the system . The basis is usually chosen such that , and is then called an orthonormal basis. In this case, the numbers , called the Fourier coefficients of the element relative to the orthonormal basis , take the form . A necessary and sufficient condition for an orthonormal system to be a basis is the Parseval–Steklov equality

for any . A Hilbert space which has an orthonormal basis is separable and, conversely, in any separable Hilbert space an orthonormal basis exists. If an arbitrary system of numbers is given such that , then in the case of a Hilbert space with a basis , the series converges in norm to an element . An isomorphism between any separable Hilbert space and the space is established in this way (Riesz–Fischer theorem).

References

[1]	L.A. Lyusternik, V.I. Sobolev, "Elements of functional analysis" , Wiley & Hindustan Publ. Comp. (1974) (Translated from Russian)
[2]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Pitman (1981) (Translated from Russian)
[3]	N.I. [N.I. Akhiezer] Achieser, I.M. [I.M. Glaz'man] Glasman, "Theorie der linearen Operatoren im Hilbert Raum" , Akademie Verlag (1958) (Translated from Russian)

References

[a1]	K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5