Categories
  Encyclosphere.org ENCYCLOREADER
  supported by EncyclosphereKSF

Bs space

From HandWiki - Reading time: 2 min


In the mathematical field of functional analysis, the space bs consists of all infinite sequences (xi) of real numbers \R or complex numbers \Complex such that supn|i=1nxi| is finite. The set of such sequences forms a normed space with the vector space operations defined componentwise, and the norm given by xbs=supn|i=1nxi|.

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

The space of all sequences (xi) such that the series i=1xi 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 via the mapping T(x1,x2,)=(x1,x1+x2,x1+x2+x3,).

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

See also

References

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





Licensed under CC BY-SA 3.0 | Source: https://handwiki.org/wiki/Bs_space
19 views | Status: cached on July 20 2024 15:22:05
↧ Download this article as ZWI file
Encyclosphere.org EncycloReader is supported by the EncyclosphereKSF