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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.

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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 $\left|S\right|>\mathfrak c$.

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