How to show B[0,1] is not separable

**teuthid** · July 2nd 2011, 03:42 PM

I'm studying for comps, and this one has me stumped:

Show that the space of bounded functions $f:[0,1]\rightarrow \mathbb{R}$ under the sup norm is not separable.

I suspect that you'd either have to show an arbitrary dense set is uncountable or that an arbitrary countable set can't be dense, but I don't have any idea how to actually implement those strategies.

**FernandoRevilla** · July 2nd 2011, 08:47 PM

For all $x\in [0,1]$ consider the family of bounded functions $f_x(t)=\begin{Bmatrix} 1 & \mbox{ if }& t=x\\0 & \mbox{if}& t\neq x\end{matrix}$
We verify $d(f_x,f_y)=1$ for all $x\neq y$ i.e., the family $\mathcal{F}=\{B(f_x,1/2):x\in[0,1]\}$ is pairwise disjoint. Now, choose $A\subset B[0,1]$ dense and use that $[0,1]$ is non denumerable.

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