By the Banach-Mazur theorem, every separable Banach space $X$ embeds into $C([0,1])$. When $X$ is reflexive, it is not possible to find a sequence of disjointly supported, non-negative functions in any isometric image of $X$ in because this would generate a copy of $c\_0$. I am intersted in some special families of functions and (im)possibility of their containment in reflexive subspaces of $X$.


Let $X$ be a separable Hilbert space. Is it possible to embed $X$ isometrically into $C([0,1])$ in such a way that the image of $X$ contains a sequence of distinct strictly increasing functions that map 0 to 0 and 1 to 1?



Strictly increasing functions in reflexive subspaces of $C([0,1])$
This is impossible. Each such function has norm 1 and only one supporting functional (point value at 1) in $C[0,1]$ so a'fortiori in this Hilbert space. But in Hilbert space the supporting functional uniquely defines the element.


