Space of null sequences

Builds on Convergent sequence Bounded sequences ℓ∞Banach space Space of convergent sequences Schauder basis

Example

Let $c_0$ be the set of sequences $x = (x_n)_{n \ge 1}$ of real (or complex) numbers with $x_n \to 0$ , with the norm $\norm{x}_\infty = \sup_n \abs{x_n}$ . Then $c_0$ is a sequence space and a Banach space.

Check: $c_0$ is the kernel of the linear functional $x \mapsto \lim_n x_n$ on the space of convergent sequences $c$ . This functional satisfies $\abs{\lim_n x_n} \le \norm{x}_\infty$ , so it is continuous, and its kernel is a closed linear subspace of $c$ . A closed subspace of a Banach space is complete, and $c$ is a Banach space, so $c_0$ is a Banach space.

Remarks

The unit vectors $e_k$ , with $1$ in place $k$ and $0$ elsewhere, form a Schauder basis of $c_0$ : for $x \in c_0$ , $\norm{x - \sum_{k \le N} x_k e_k}_\infty = \sup_{k > N} \abs{x_k} \to 0$ . In particular $c_0$ is separable, unlike the space $\ell^\infty$ of bounded sequences. The dual space of $c_0$ is isometrically isomorphic to $\ell^1$ , through the pairing $(a, x) \mapsto \sum_n a_n x_n$ . The space $c$ is the direct sum of $c_0$ and the line of constant sequences.

Reading path

To understand this page from the ground up, read these 49 pages in order. The foundations come first.

Bounded sequence concept
Convergent sequence concept
Element definition
Extended real number line object
Proposition concept
Real numbers object
Schauder basis concept
Separable space concept
Sequence definition
Set bounded above definition
p-metric on R^n example
Set definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Real sequence definition
Supremum definition
p-norm on R^n example
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Existential quantifier definition
Universal quantifier definition
Cauchy sequence definition
Cartesian product definition
Subset definition
Union of a family of sets definition
Uniqueness quantifier definition
Binary relation definition
Topological space definition
Function definition
Metric space definition
Convergence of a sequence in a metric space definition
Open ball definition
Diameter of a set definition
Metric topology definition
Epsilon characterisation of Cauchy sequences statement
Complete metric space definition
Open set definition
Equivalent metrics definition
Equivalent norms definition
Banach space definition
Sequence spaces lp example
Bounded sequences ℓ∞ example
Space of convergent sequences example
Space of null sequences example

Tag	`WDH6QSH1Z6`
URI	`m2://WDH6QSH1Z6`
Kind	example
Writing	W0 stub
MSC	46B45, 46A45
Instance of	Sequence space, Banach space
Uses	Convergent sequence, Bounded sequences ℓ∞, Banach space, Space of convergent sequences, Schauder basis
Changed	2 November 2026

Space of null sequences

Example

Remarks

Reading path

Further reading

What this page needs for W1

What links here