Analysis Problem of the Day 35

Today’s problem appears as Problem 4 on Texas A&M’s August 2017 Real Analysis Qual:

Problem 4: a) Prove or disprove that $L^\infty([0,1])$ is separable.

b) Find, with proof, the weak-* closure of the unit ball of $C([0,1])$ in $L^\infty([0,1]).$

Solution: a) It is a standard fact that $L^p$ is separable if and only if $p<\infty.$ We recall the proof of non-separability for $p=\infty:$ Consider the uncountable set of elements $a_r(x):=\chi_{[0,r]}(x), r \in [0,1],$ and note that $\|a_r-a_{r'}\|=2\delta_{r,r'}.$ This shows that there are uncountably many disjoint balls of radius 1, so any dense subset would have to have a point in each one of those balls and therefore be at least uncountable. Thus, $L^\infty([0,1])$ is not separable.

b) I claim that the weak-* closure of the unit ball of $C([0,1])$ in $L^\infty([0,1])$ is the unit ball in $L^\infty([0,1]).$ Since $L^\infty = (L^1)^*$ and $L^1$ is separable, the unit ball in $L^\infty$ is weak-* metrizable, so it suffices to deal with sequential convergence. We claim that for all $f \in L^\infty([0,1]), \|f\|_\infty \leq 1,$ there exists a sequence $g_n \in C([0,1]), \|g_n\|_\infty \leq 1,$ such that $g_n \overset{*}{\rightharpoonup} f,$ i.e. $\int_0^1 g_n h \to \int_0^1 f h$ for all $h \in L^1([0,1]).$ Indeed, by density of continuous functions in $L^1,$ take $h_n \in C([0,1]), h_n \to f,$ and let $g_n:=\max(-\|f\|_\infty,\min(h_n,\|f\|_\infty)).$ By Egorov’s theorem, $g_n$ converges uniformly to $f$ on $K$ with $\mu(K^c)<\delta,$ since for any point $x \in K,$ $h_n(x) \to f(x)$ and $|f(x)| \leq \|f\|_\infty$ implies $g_n(x) \to f(x).$ Meanwhile, on $K^c,$ $\|g_n\|_\infty \leq \|f\|_\infty,$ so picking $\delta$ small enough so that $\|h\|_{L^1(K^c)}<\epsilon,$ by Hölder’s,

$\left|\int_0^1 (g_n-f)h \right|\leq \epsilon \|h\|_1 + \|g_n-f\|_\infty \|h\|_{L^1(K^c)} \leq \epsilon (\|h\|_1+2\|f\|_\infty)$

for large enough $n.$ Thus, $g_n$ converges to $f$ weak-* in $L^\infty,$ so the closure of the unit ball of $C([0,1])$ is the unit ball in $L^\infty([0,1]).$

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