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 $X$ 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?

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?