[Math] Nonseparable counterexamples in analysis

Question

When asking for uncountable counterexamples in algebra I noted that in functional analysis there are many examples of things that “go wrong” in the nonseparable setting. But most of the examples I'm thinking of come from $\mathrm C^*$-algebras (equivalent formulations of being type I; anything having to do with direct integral decompositions; primeness versus primitivity, etc.).

Question. Are there also good examples from Banach space theory, or from harmonic analysis? Or from analysis in general? (Or even other
examples from $\mathrm C^*$-algebra that I might not know about?)

Answer 1

Sobczyk's theorem: if $Z$ is a subspace of a separable Banach space $X$ that is isomorphic to $c_0$, then $Z$ is complemented in $X$, fails for many non-separable spaces such as $X=\ell_\infty\cong C( \beta \mathbb N)$.

For related reasons, the Borsuk–Dugundji extension theorem (which says that if $F$ is a closed, metrisable subspace of a compact space $K$, then you can apply the Tietze–Urysohn extension theorem in a linear way, i.e., there is a contractive operator $T\colon C(F)\to C(K)$ such that $(Tf)|_F=f$ for $f\in C(F)$) fails when $F$ is non-metrisable, that is, $C(F)$ is non-separable in the uniform norm. (Here, $F = \beta \mathbb N\setminus \mathbb N \subset \beta\mathbb N$ is the easiest counterexample.)