Sequence space
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In functional analysis and related areas of mathematics, a sequence space is an important class of function space.
The set of all functions from the natural numbers to complex numbers, which can naturally be identified with the set of all possible infinite sequences with elements in 
, can be turned into a vector space. Any linear subspace of this space is then called sequence space.
Many important classes of sequences like bounded sequences or null sequences form sequence spaces. A sequence space equipped with the topology of pointwise convergence becomes a special kind of Fréchet space called FK-space.
[edit] Definition
We identify the set of all functions
with the set of all sequences
with 
This set can be turned into a vector space by defining vector addition as
and the scalar multiplication as
A sequence space X is a linear subspace of ω.
[edit] Examples
The space of bounded sequence 
(sometimes called m) consisting of all bounded sequences
The space of convergent sequences c consisting of all convergent sequences
The space of null sequences c0 consisting of all null sequences
The space of finite sequences Φ consisting of all sequences where only a finite number of terms are non-zero.
The space of bounded series bs
[edit] See also
- Lp space
- FK-space, sequence spaces which are also Fréchet space
- beta-dual space
