Banach space with cardinality bigger than $\mathfrak{c}$.

Question

By using the information contained in this post, we have that the cardinality of every Banach space is equal to its dimension, which in turn, is bigger or equal to $\mathfrak{c}$.

In my area of study, I have always beem studying spaces like $W^{1,p}$ for $p\in (1,\infty)$ which are separable. Moreover, the space that I know which are not separable is $L^\infty$, but I don't know how to calculate its cardinality.

My question is: Is there a example of a Banach space with cardinality bigger than $\mathfrak{c}$?

Remark: I'm not used to study those things, hence, if I have posted something stupid here, please neglect it.

Answer 1

Let $S$ be any set, and $\ell^\infty(S)$ the space of bounded maps $S \to \mathbb R$, with the norm $$ \|x\|_{\ell^\infty(S)} := \sup_{s\in S}\left|x(s)\right| $$ It is complete and $\left|\ell^{\infty}(S)\right| \ge \left|S\right|$. Now choose $S$ with $\left|S\right|>\mathfrak c$.