TITLE: Would you recommend me a text book reference?
QUESTION [7 upvotes]: I am searching for a text book reference or lecture notes, where I can find the following theorem:
I mean as similar as possible to the statement with these same words (if possible). Or including these 3 equivalences:
THEOREM: Let $M\subset H$, orthonormal, are equivalents
a) $M$ is total basis (or orthonormal basis or Hilbert basis);
b) For all $u \in H\quad  \sum_{i\in Ju}$, $(u, u_i)u_i=u$ where $Ju=\{i\in J: u_i\in M \; \text{and} \; (u_i, u)\neq 0\}$
c) $\sum_{i\in J_u} |(u, u_i)|^2=|u|^2$, for all $u \in H$ (Parseval identity)
Remark: I wish this because I'm studying using Brezis' book, however there is appearing in different theorems, and it is not including the $\iff$ (equivalence).

REPLY [2 votes]: A Bit more of that theorem can be found in the page 108 of Fourier and Wavelet Analysis By Bachmann, Narici, Beckenstein.

or in the page 346 of
Roman, Steven, Advanced linear algebra, Graduate Texts in Mathematics. 135. New York: Springer-Verlag. xii, 363 p. (1992). ZBL0754.15002.