The Banach-Mazur theorem says that every separable Banach space is isometric to a subspace of $C^0([0;1],R)$, the space of continuous real valued functions on the interval $[0;1]$, with the sup norm.


If we apply this to $\ell^2(R)$, then we see that $C^0([0;1],R)$ has a subspace which is a Hilbert space for the sup norm. 


My question is can one write down explicitly such a subspace of $C^0([0;1],R)$?


I'm just curious, that's all.



Banach-Mazur applied to a Hilbert space
Let $\varphi$ be a continuous function from $[0,1]$ onto the closed unit ball $B$ of $\ell^2$ in the weak topology. Then we define $J: \ell^2 \to C[0,1]$ by $(Jx)(t) = <\phi(t), x>$. Now, how to construct $\varphi$? 


Consider convex compact subsets of $B$ of the form $K(a\_1,\ldots,a\_n) = \{x \in B: a\_i/2^n \le x\_i \le (a\_i+1)/2^n \text{ for } i = 1 \ldots n\}$ for integers $a\_i$, $-2^n \le a\_i \le 2^n$. Of course many of these are empty, and we disregard those. We can consider the nonempty ones as forming a 
tree structure $T$ with vertices $v = (a\_1, \ldots, a\_n)$, and edges joining $(a\_1, \ldots, a\_n)$ to its children $(b\_1, \ldots, b\_{n+1})$ where for $1 \le i \le n$, $b\_i$ is either $2 a\_i$ or $2 a\_i + 1$. Thus $K(v) = \bigcup\_w K(w)$ where $w$ runs over the children of $v$. The root $r$ of the tree corresponds to $B$ itself (with $n=0$).


Recursively define closed subintervals $J(v)$ of $J(r) = [0,1]$ for $v \in T$ so that


1. If $w$ is a child of $v$, $J(w) \subset \text{interior}(J(v))$
2. If $w\_1$ and $w\_2$ are disjoint children of $v$, $J(w\_1)$ and $J(w\_2)$ are disjoint.
3. If $v$ is at level $n$ in the tree, $J(v)$ has length at most $2^{-n}$.


Let $E\_n$ be the union of $J(v)$ for all vertices $v$ at level $n$.
Define $\varphi\_n: [0,1] \to B$ by selecting $x\_v \in K(v)$ for each vertex $v$ at level $n$, and defining $\varphi\_n(t) = \varphi\_{n-1}(t)$ for $t \notin \text{interior}(E\_n)$,
$\varphi\_n(t) = x\_v$ for $t \in J(v)$ where $v$ is a vertex at level $n$,
and interpolating linearly on the rest. Note that if $m > n$ and $t \in J(v)$ where $v$ is a vertex at level $n$, $\varphi\_m(t) \in K(v)$. 


Finally, define $\varphi(t) = \lim\_{n \to \infty} \varphi\_n(t)$. This, I claim, is 
a continuous surjective function from $[0,1]$ to $B$ with the weak topology.


