Sequence space

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In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of complex numbers. Equivalently, it is function space whose elements are functions from the natural numbers to the complex numbers.

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 \mathbb{C}, can be turned into a vector space. All sequence spaces are linear subspaces of this 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, \mathbb{C}^{\mathbb{N}}, of all functions

f:\mathbb{N} \to \mathbb{C}

with the set of all sequences

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

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

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

and the scalar multiplication as

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

A sequence space, X, is a linear subspace of \mathbb{C}^{\mathbb{N}}.

[edit] Examples

The space of bounded sequences \ell^\infty (sometimes called m)

m:=\{ x \in \mathbb{C}^{\mathbb{N}} : \exists M \in \mathbb{R} \quad \forall n \in \mathbb{N} \quad \vert x_n \vert \le M\}.

The space \Phi \! of all infinite sequences with only a finite number of non-zero terms (sequences with finite support).

The Lp sequence spaces \ell^p, which consists of sequences such that the p-power norm is finite:

\|x\|_p=\left( \sum_i |x_i|^p \right)^{1/p} < \infty

The space of convergent sequences c

c:=\{ x \in \mathbb{C}^{\mathbb{N}} : \exists M \in \mathbb{C} \quad \lim_{n\to\infty} (x_n - M) = 0\}.

The space of null sequences c0

c_0:=\{ x \in \mathbb{C}^{\mathbb{N}} : \lim_{n\to\infty} x_n = 0\}.

The space of bounded series bs

bs:=\{x \in \mathbb{C}^{\mathbb{N}} : \sup_n \vert \sum_{i=0}^n x_i \vert < \infty \}.

[edit] See also

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