information:filtering
Second example: sequence spaces

The sequence space l consists of all infinite sequences z = (z, z, …) of complex numbers such that the following series converges: ∑ n = 1 ∞ | z n | 2 {\displaystyle \sum _{n=1}^{\infty }|z_{n}|^{2}}

The inner product on l is defined by: ⟨ z, w ⟩ = ∑ n = 1 ∞ z n w n ¯, {\displaystyle \langle \mathbf {z},\mathbf {w} \rangle =\sum _{n=1}^{\infty }z_{n}{\overline {w_{n}}}\,,}

This second series converges as a consequence of the Cauchy–Schwarz inequality and the convergence of the previous series.

Completeness of the space holds provided that whenever a series of elements from l converges absolutely (in norm), then it converges to an element of l. The proof is basic in mathematical analysis, and permits mathematical series of elements of the space to be manipulated with the same ease as series of complex numbers (or vectors in a finite-dimensional Euclidean space).