Let $t\ge 0$ and suppose some Hilbert space $H$ is isometrically isomorphic to $C_F[0,t]$ (continuous maps $[0,t] \to F$, with the supremum norm) where $F=\mathbb{C}$ or $\mathbb{R}$. I am asked to determine $t$ and $H$ up to isometric isomorphism.
I don't know much about Hilbert spaces, and the only thing I can find in my book is that an infinite-dimensional separable Hilbert space over $F$ is isometrically isomorphic to $\ell^2_F$. I'm not seeing how/if this is useful.
Any help would be appreciated!
Here's a hint: If $t>0$, it's not too hard to find two functions $f,g\in C_F([0,t])$ such that $\|f\|=\|g\|=\|f+g\|=1$. Show that this is impossible in Hilbert space.