Orthogonal basis

A system of pairwise orthogonal non-zero elements $e_1,\dots,e_n,\dots,$ of a Hilbert space $X$, such that any element $x\in X$ can be (uniquely) represented in the form of a norm-convergent series

$$x=\sum_ic_ie_i,$$

called the Fourier series of the element $x$ with respect to the system $\{e_i\}$. The basis $\{e_i\}$ is usually chosen such that $\|e_i\|=1$, and is then called an orthonormal basis. In this case, the numbers $c_i$, called the Fourier coefficients of the element $x$ relative to the orthonormal basis $\{e_i\}$, take the form $c_i=(x,e_i)$. A necessary and sufficient condition for an orthonormal system $\{e_i\}$ to be a basis is the Parseval–Steklov equality

$$\sum_i|(x,e_i)|^2=\|x\|^2,$$

for any $x\in X$. 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 $\{c_i\}$ is given such that $\sum_i|c_i|^2<\infty$, then in the case of a Hilbert space with a basis $\{e_i\}$, the series $\sum_ic_ie_i$ converges in norm to an element $x\in X$. An isomorphism between any separable Hilbert space and the space $l_2$ 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