Bs space

Template:Short description In the mathematical field of functional analysis, the space bs consists of all infinite sequences (x_i) of real numbers $ℝ$ or complex numbers $ℂ$ such that $$\underset{n}{\sup }\left|\sum _{i=1}^n{x}_i\right|$$ is finite. The set of such sequences forms a normed space with the vector space operations defined componentwise, and the norm given by $$\Vert x{\Vert }_{bs}=\underset{n}{\sup }\left|\sum _{i=1}^n{x}_i\right|.$$

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 $$\sum _{i=1}^{\mathrm{\infty }}{x}_i$$ 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.

The space bs is isometrically isomorphic to the Space of bounded sequences ${\ell }^{\mathrm{\infty }}$ via the mapping $$T(x_1,x_2,\dots )=(x_1,x_1+x_2,x_1+x_2+x_3,\dots ).$$

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

See also

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References

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