TITLE: Why is Parseval's Equality and Bessel's Inequality Different?
QUESTION [5 upvotes]: Bessel's Inequality:   $\sum_n |\langle x, e_n \rangle |^2 \leq \|x\|^2$
Parseval: $\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ $\sum_n |\langle x, e_n \rangle |^2 = \|x\|^2$

REPLY [13 votes]: You forgot to say what are you summing on. Bessel works for any orthonormal sequence; for Parseval to work the orthonormal sequence must be complete.

REPLY [3 votes]: Parseval's identity holds in any inner-product space, not just separable Hilbert spaces.  Bessel's inequality is a statement about the coefficients of an element  in a Hilbert space with respect to an orthonormal sequence as pointed out by @Gadi A