The space of continuous function from an interval into a Hilbert space is homeomorphic with the Hilbert space

Question

First, I would like to thank the organizers as well as the people who participate in this forum Second, Let $H$ be an infinite dimensional Hilbert space and $I$ a closed interval, $C\left(I,H\right)$ is the space of Continuous function My question is as follows: Is the space of Continuous functions $C\left(I,H\right)$ is homeomorphic to $H$?

Answer 1

Theorem 6.1 of this paper implies homoemorphism classes of Banach spaces are determined entirely by their density character. For each cardinality $\alpha$ there is a unique Hilbert space $\ell^2(\alpha)$ for which $\alpha$ is the density character. So each Banach space $V$ is topologically equivalent to $\ell^2(\alpha)$ for $\alpha$ the density of $V$.

This answer shows the special case that $C(I,\ell^2(\mathbb N))$ is separable hence homeomorphic to $\ell^2(\mathbb N)$. The proof for the more general $\ell^2(\alpha)$ should be almost identical.