Let $\{e_k\}_k$ be a basis for $ L^2(X,\mu)$, and $\{a_k\}_k$ limited sequence of real numbers.
$X$ is a compact metric space, and $\mu$ is the borelian measure.
Can one say the set $\{g \in L^2(X,\mu) / g = \displaystyle\sum_{i=1}^\infty b_ke_k $ and $|b_k|\leq |a_k|\}$ is homeomorphic to the Hilbert Cube right away, or more caution is demanded?
Thanks in advance