Show that every Banach space is isometric to a subspace of an $L_{\infty}(\mu)$-space

Question

The following statement can be obtain in the proof of Proposition $7.1$

Since every Banach space is isometric to a subspace of an $L_{\infty}(\mu)$-space (e.g. an $l_{\infty}(T)$ for suitable $T$), the proposition is trivial if $p=\infty$.

Question: How to show that 'every Banach space is isometric to a subspace of an $L_{\infty}(\mu)$-space'?

Answer 1

Let $X$ be a Banach space and let $B_{X^*}$ be the unit ball of $X^*$. Define a map $T\colon X\to \ell_\infty(B_{X^*})$ by $$Tx = (\langle f, x\rangle)_{f\in B_{X^*}}\quad (x\in X).$$ Then $T$ is linear. It follows from the Hahn–Banach theorem that $T$ is isometric.

In some sense the answer is optimal because spaces which do not admit a strictly convex renorming (such as $\ell_\infty/c_0$) do not embed into $L_\infty(\mu)$ for $\mu$ being $\sigma$-finite.