Bs space explained

\R or complex numbers \Complex such that$$\backslash sup\_n\; \backslash left|\backslash sum\_^n\; x\_i\backslash right|$$is finite. The set of such sequences forms a normed space with the vector space operations defined componentwise, and the norm given by$$\backslash |x\backslash |\_\; =\; \backslash sup\_n\; \backslash left|\backslash sum\_^n\; x\_i\backslash right|.$$

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

The space of all sequences

\left(x _{i\right)} such that the series$$\backslash sum\_^\backslash infty\; x\_i$$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.\ell ^{infty} via the mapping$$T(x\_1,\; x\_2,\; \backslash ldots)\; =\; (x\_1,\; x\_1+x\_2,\; x\_1+x\_2+x\_3,\; \backslash ldots).$$

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

T.

References

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