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Square-Summable Sequence

A square-summable sequence is a sequence (u_n) of real or complex numbers for which

sum_(n)|u_n|^2<infty.

The index may run over the positive integers or all integers. Such a sequence is an element of l^2, the L2-space for counting measure, and has l2-norm

||u||_2=(sum_(n)|u_n|^2)^(1/2).

The space l^2 with this norm is a Hilbert space (Axler 2026, pp. 194-195 and 224).

See also

Hilbert Space, L2-Norm, L2-Space, Sequence

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References

Axler, S. Measure, Integration & Real Analysis. Author-hosted revision, June 12, 2026. https://measure.axler.net/MIRA.pdf.

Cite this as:

Weisstein, Eric W. "Square-Summable Sequence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Square-SummableSequence.html

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