Definition of the spaces $c_{00}$ and $c_0$

definitionfunctional-analysisreference-request

I am given the following chain of subspaces:

Let $S$ be a set. Then we have $c_{00}(S)\subseteq\ell^1(S)\subset c_0(S)\subset \ell^\infty(S)$.
The context is reflexivity and the following theorem is proven:

There is an isometric isomorphism $\Phi:\ell^1(S)\to c_0(S)^\ast$ given by $\Phi(f)(g)=\sum_{s\in S} f(s)g(s)$ for all $f\in\ell^1(S)$ and $g\in c_0(S)$.

I have tried to find the notation in several books on functional analysis.
$c_0(S)$ seems to be the set of sequences converging to $0$.

And $c_{00}(S)$ should be the set of convergent sequences.

Can you confirm this, or give a reference?
I have also looked at this question: $c_0, \ell^1,\ell^\infty$ and their Dual Spaces: Rudin's RCA, Problem $5.9$

And tried to find the notation in Rudin's Book, but it is not listed in the symbol reference.

Thanks in advance.

Best Answer

Firstly, you might need more structure on the set $S$. It would be nice if $S$ were a normed-linear space, and nicer if it were Banach. For the time being, allow me to assume that $\|\cdot\|$ is a norm on $S$.

  1. $c_{00}(S)$ refers to the space of sequences with finitely many non-zero terms. In other words, if $\{x_n\}_{n=1}^\infty \in c_{00}(S)$, then there exists $N\in \mathbb N$ such that $x_n = 0$ for all $n\ge N$.

  2. $c_{0}(S)$ refers to the space of sequences that converge to zero. That is, if $\{x_n\}_{n=1}^\infty \in c_{0}(S)$, then $x_n \xrightarrow{n\to\infty} 0$ in the topology induced by the norm on $S$.

  3. $\ell^1(S)$ consists of sequences $\{x_n\}_{n=1}^\infty \subset S$ satisfying $$\sum_{n=1}^\infty \|x_n\| < \infty$$

  4. Lastly, $\ell^\infty(S)$ consists of sequences $\{x_n\}_{n=1}^\infty \subset S$ satisfying $$\sup_{n\ge 1} \|x_n\| < \infty$$

Exercise. Suppose $S = \mathbb C$ with the usual norm. Can you see which inclusions are strict?

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