# Bs space

In the mathematical field of functional analysis, the space **bs** consists of all infinite sequences (*x*_{i}) of real or complex numbers such that

is finite. The set of such sequences forms a normed space with the vector space operations defined componentwise, and the norm given by

Furthermore, with respect to metric induced by this norm, *bs* is complete: it is a Banach space.

The space of all sequences (*x*_{i}) such that the series

is convergent (possibly conditionally) is denoted by *cs*. This is a closed vector subspace of *bs*, and so is also a Banach space with the same norm.

The space *bs* is isometrically isomorphic to the space of bounded sequences ℓ^{∞} via the mapping

Furthermore, the space of convergent sequences *c* is the image of *cs* under *T*.

## References

- Dunford, N.; Schwartz, J.T. (1958),
*Linear operators, Part I*, Wiley-Interscience.

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