Timeline for Non-separable Banach space

10 events

when toggle format	what		by	license	comment
Mar 23, 2015 at 9:58	answer	added	Ian Morris		timeline score: 2
Feb 11, 2015 at 9:40	comment	added	Ian Morris		For what it's worth (if anything), I wouldn't prove $L^\infty(\mathbb{R})$ to be non-separable in the way you suggest: I'd instead do it by observing that the subset $\{\chi_{(-\infty,t)}\colon t \in \mathbb{R}\}$ is isometric to $\mathbb{R}$ with the discrete metric.
Feb 10, 2015 at 21:22	answer	added	weather		timeline score: 3
Feb 10, 2015 at 19:25	comment	added	Nate Eldredge		This reminds me a little of this question. One sophisticated way to prove this might be that $C_b(\mathbb{R})$ is isometrically isomorphic to $C(\beta \mathbb{R})$ where $\beta \mathbb{R}$ is the Stone-Cech compactification. If $C(\beta \mathbb{R})$ were separable then $\beta \mathbb{R}$ would be second countable (and in fact metrizable), but that isn't true.
Feb 10, 2015 at 18:33	history	edited	Ricardo Andrade	CC BY-SA 3.0	edited tags
Feb 10, 2015 at 15:06	comment	added	Ian Morris		What would be an example of a non-direct proof?
Feb 10, 2015 at 12:57	answer	added	Fedor Petrov		timeline score: 10
Feb 10, 2015 at 12:33	comment	added	Pietro Majer		For instance the discrete uncountable metric space $P(\mathbb{N})$ endowed with the Hausdorff distance, embeds isometrically into $C_b(\mathbb{R})$ via $S\mapsto v_S$ s.t. $v_S(x):=\mathrm{dist}_S(x)$.
Feb 10, 2015 at 9:52	answer	added	Johannes Hahn		timeline score: 8
Feb 10, 2015 at 9:32	history	asked	Bazin	CC BY-SA 3.0