From Wikipedia, the free encyclopedia
In the
mathematical field of
functional analysis, the space bs consists of all infinite
sequences (xi) of
real numbers
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
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
via the mapping
Furthermore, the
space of convergent sequences
c is the image of cs under
See also
References
- Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.
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