I would like to ask about a problem I met:
If $U$ and $H$ are two real Hilbert spaces (they are infinite dimensional), let $C_{m}(U,H)$ denote the space of continuous maps from $U$ to $H$ which have at most polynomial growth of order $m$. It is known that for any $u \in U$, if we have $|u|_{C_{m}}$= $sup_{h \in H} \frac {|u(h)|} {1+|h|^{m}}$, then $C_{m}(U,H)$ is Banach space with this norm.
I would like to know
1. Is $C_{m}(U,H)$ also separable?
2. If not, is it possible to give it a norm so that $C_{m}(U,H)$ is a separable Banach space?
Thank you very much!
Is the space of maps between Hilbert spaces that have at most polynomial growth of order m a separable Banach space
functional-analysisreal-analysis
Best Answer
No and no. First, one can not ask if $C_{m}(U,H)$ is separable until it has some topology on it, e.g. the one defined by the norm. Second, even the space of bounded linear operators, i.e. continuous linear polynomials, on a separable Hilbert space is non-separable. It contains an isometric copy of $\ell_\infty$ represented by the "diagonal" operators, see Separability of the space of bounded operators on a Hilbert space. Spaces of continuous maps between two infinite-dimensional spaces are just too large to expect separability.
Bounded operators can be approximated by finite rank ones in weak and strong operator topologies, see Weak* operator topology and finite rank operators, but one does not usually talk about separability in these topologies. One can build separable spaces of polynomials by taking various closures of polynomials built from finite rank linear monomials, see e.g. the Hilbert space of Wick polynomials from white noise analysis. But they will be far smaller than all continuous maps of polynomial growth.
Nemirovskiĭ and Semenov studied approximation of uniformly continuous functions on the unit ball of a separable Hilbert space in On Polynomial Approximation of Functions on Hilbert Space. They give an example of a function having uniformly continuous derivatives of all orders but no polynomial approximation, and that's on a ball. There is nothing countable that one can build countable basis from in such spaces.