What is the cardinality of $L^{\infty}(\mathbb{R})$?

Question

I know of this answer for $L^p(\mathbb{R})$ where $p < \infty$: What is the cardinality of $L^p(\mathbb R)$, $1 \le p < \infty$?

But, what about $L^{\infty}(\mathbb{R})$, can we prove any result on its cardinality? I know there are non-separable Hilbert spaces which can have arbitrary large cardinalities.

Answer 1

$|L^\infty(\Bbb R)|=c=|\Bbb R|$:

Since $t\mapsto\chi_{(t,\infty)}$ gives an injective map from $\Bbb R$ to $L^\infty(\Bbb R)$ it's clear that $|L^\infty(\Bbb R)|\ge c$.

Since $$L^\infty(\Bbb R)=\prod_{n\in\Bbb Z}L^\infty([n,n+1))$$and $$c^{\aleph_0}=2^{\aleph_0\aleph_0}=c,$$it's enough to show $|L^\infty([0,1))|\le c$. But $L^\infty([0,1))\subset L^2([0,1))$.