Space of bounded sequences
In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences of real numbers or complex numbers. When equipped with the uniform norm:
the space
becomes a
Banach space. It is a
closed linear subspace of the
space of bounded sequences,
, and contains as a closed subspace the Banach space
of sequences converging to zero. The
dual of
is isometrically isomorphic to
as is that of
In particular, neither
nor
is
reflexive.
In the first case, the isomorphism of with is given as follows. If then the pairing with an element in is given by
This is the Riesz representation theorem on the ordinal .
For the pairing between in and in is given by
See also[edit]
References[edit]
- Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.
|
---|
Spaces | |
---|
Theorems | |
---|
Operators | |
---|
Algebras | |
---|
Open problems | |
---|
Applications | |
---|
Advanced topics | |
---|
|