Sequence space

Remarks

Example: the set $\ell^\infty$ of bounded sequences is a sequence space. It contains the zero sequence, and if $\abs{x_n} \le M$ and $\abs{y_n} \le N$ for all $n$, then $\abs{x_n + y_n} \le M + N$ and $\abs{\lambda x_n} \le \abs{\lambda} M$. Other examples are $\omega$ itself, the spaces ℓ^p, the space $c$ of convergent sequences and the space $c_0$ of null sequences. Another is the space $\varphi$ of sequences with only finitely many nonzero terms.

Non-example: the set of sequences with $x_1 = 1$ is not a sequence space, because it does not contain the zero sequence.

The word is often used for a sequence space together with a norm or a topology, as in the theory of FK-spaces and BK-spaces.