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 <math>\mathbb{C}<math>, 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.

Definition

We identify the set of all functions

<math>f:\mathbb{N} \to \mathbb{C}<math>

with the set of all sequences

<math>(x_n)_{n\in\mathbb{N}}<math> with <math>x_n \in \mathbb{C}.<math>

This set can be turned into a vector space by defining vector addition as

<math>(x_n)_{n\in\mathbb{N}} + (y_n)_{n\in\mathbb{N}} := (x_n + y_n)_{n\in\mathbb{N}}<math>

<math>\alpha(x_n)_{n\in\mathbb{N}} := (\alpha x_n)_{n\in\mathbb{N}}.<math>

A sequence space <math>X<math> is a linear subspace of <math>\omega.<math>

The space of bounded sequence <math>l^\infty<math> (sometimes called <math>m<math>) consisting of all bounded sequences

<math>m:=\{ x \in \omega : \exists M \in \mathbb{R} \quad \forall n \in \mathbb{N} \quad \vert x_n \vert \le M\}.<math>

The space of convergent sequences <math>c<math> consisting of all convergent sequences

<math>c:=\{ x \in \omega : \exists M \in \mathbb{C} \quad \lim_{n\to\infty} (x_n – M) = 0\}.<math>

The space of null sequences <math>c_0<math> consisting of all null sequences

<math>c_0:=\{ x \in \omega : \lim_{n\to\infty} x_n = 0\}.<math>

The space of finite sequences <math>\Phi<math> consisting of all sequences where only a finite number of terms a non-zero.

The space of bounded series <math>bs<math>

<math>bs:=\{x \in \omega : \sup_n \vert \sum_{i=0}^n x_i \vert < \infty \}.<math>