Hilbert space

This property expresses the completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense.

This last property is ultimately a consequence of the more fundamental Cauchy–Schwarz inequality, which asserts

with the latter series converging as a consequence of the Cauchy–Schwarz inequality and the convergence of the previous series.

Superposition of sinusoidal wave basis functions (bottom) to form a sawtooth wave (top)

There is also a version of the spectral theorem that applies to unbounded normal operators.