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Bessel's Inequality: $\sum_n |\langle x, e_n \rangle |^2 \leq \|x\|^2$

Parseval: $\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ $\sum_n |\langle x, e_n \rangle |^2 = \|x\|^2$

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2 Answers 2

up vote 2 down vote accepted

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

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You forgot to say what are you summing on. Bessel works for any othrthonormal sequence; for Parseval to work the orthonormal sequence must be complete.

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