Closed subspaces of $L^{\infty}[0,1]$

Question

I would like to prove that $L^\infty[0,1]$ (bounded functions in $[0,1]$) has closed subspaces isomorphic to $c_0$ (space of sequences converging to zero). Do you have any ideas? Thanks in advance.

Answer 1

Consider the subspace $Y$ of $L^\infty[0,1]$ consisting of functions of the form $f = \sum_{n=1}^\infty \alpha_n \mathbb{1}_{(\frac{1}{n+1},\frac{1}{n}]}$ where $\alpha_n \in \ell^\infty$.

It is clear that $Y$ is isometrically isomorphic to $\ell^\infty$ via the map $(\alpha_n) \mapsto \sum_{n=1}^\infty \alpha_n \mathbb{1}_{(\frac{1}{n+1},\frac{1}{n}]}$. But then since $c_0$ is a closed subspace of $\ell^\infty$ it is isometrically isomorphic to a closed subspace of $Y$ (namely the image of $c_0$ under this map) and hence of $L^\infty[0,1]$.