Sequence space

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.

Table of contents

1 Definition 2 Examples 3 See also

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 XX is a linear subspace of

Examples

The space of bounded sequence (sometimes called mm) consisting of all bounded sequences

The space of convergent sequences cc consisting of all convergent sequences

The space of null sequences consisting of all null sequences

The space of finite sequences consisting of all sequences where only a finite number of terms a non-zero.

The space of bounded series bsbs

See also

• FK-space, sequence spaces which are also Fréchet space
• beta-dual space

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