Let $S$ be an uncountable set. Consider the subspace $\ell\_\infty(\kappa, S)$ of $\ell\_\infty(S)$ formed by all functions with support of cardinality at most $\kappa$ (here $\kappa<|S|$). Certainly, $\ell\_\infty\subset \ell\_\infty(\kappa, S)$, whence the density character of $\ell\_\infty(\kappa, S)$ is at least $2^\omega$. Can we specify it? Is it for example $2^\omega \cdot \kappa$?



Density character of $\ell_\infty(\kappa, S)$
I understand you to mean that $\ell\_\infty(S)$ is the collection of
bounded functions from $S\to\mathbb{R}$, under the sup norm, and
$\ell\_\infty(\kappa,S)$ consists of those functions which take value
$0$ outside a set of size at most $\kappa$, a fixed infinite
cardinal less than $|S|$. The density character is the size of the
smallest dense subset of this space under the sup norm metric.


I claim that the density character is exactly $|S|^\kappa$. It can
be no larger than this, because the number of possible supports is
$|S|^\kappa$, and for each support, there are
$|\mathbb{R}|^\kappa=2^\kappa$ many functions having that support,
leading to $|S|^\kappa\cdot 2^\kappa=|S|^\kappa$ many functions
altogether in $\ell\_\infty(\kappa,S)$. Conversely, the density
character is at least this large. To see this, consider the set of
all functions $f:S\to
\{0,1\}$ with support of size $\kappa$. There are precisely
$|S|^\kappa$ many such functions, and any two of these functions
have distance $1$. So any dense set must have elements close to
each of these functions, and so must have size at least
$|S|^\kappa$.


Regarding the $2^\omega\cdot\kappa$ proposal, note that $|S|^\kappa$ is at least $2^\kappa$, which is strictly larger than $2^\omega\cdot\kappa$ when $\kappa\geq 2^\omega$.


(Lastly, if I may, allow me to suggest a small modification to
your notation. It might be better to consider $\kappa\leq|S|$ and
define that $\ell\_\infty(\kappa,S)$, or perhaps denote it $\ell\_\infty({\lt}\kappa,S)$, to be the collection of functions
in $\ell\_\infty(S)$ with support of size *strictly less than*
$\kappa$, rather than at less-than-or-equal-to $\kappa$. This way
of doing it includes your case, by replacing $\kappa$ with
$\kappa^+$, but also allows one to consider the case of limit
cardinals, such as the functions with support less than
$\aleph\_\omega$, which is not possible to express in your
notation. A similar argument to the above shows that $\ell\_\infty({\lt}\delta,S)$ has density character $|S|^{\lt\delta}$)


