Theorem 14.4.3 (Banach-Mazur).label Let $E$ be a separable normed vector space over $K \in \RC$, then there exists an isometric embedding $\iota \in L(E; C([0, 1]; K))$.
Proof. Let $B$ be the closed unit ball of $E^{*}$, equipped with the weak* topology. By the Hahn-Banach Theorem, the linear mapping
is an isometric embedding. By Proposition 14.4.1, $B$ is a compact metric space. The Alexandroff-Hausdorff Theorem then provides a continuous surjection $f: 2^{\natp}\to B$. Thus the composition map
is a linear isometric embedding. Let $\mathcal{C}\subset [0, 1]$ be the Cantor set, then $\mathcal{C}$ is homeomorphic to $2^{\natp}$ through Proposition 10.3.4. Hence $C(2^{\natp}; K)$ is isometrically isomorphic to $C(\mathcal{C}; K)$.
Finally, Lemma 14.4.2 provides yet another linear isometric embedding $C(\mathcal{C}; K) \to C([0, 1]; K)$. Composing the above maps as follows
yields the desired embedding.$\square$
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