sg.symplectic geometry - Floer's space closed under products?

Floer chooses $epsilon_k$ so that this $epsilon$ space is dense in $L^2$ (this should be equivalent to dense in $C^infty$, since the latter is dense in $L^2$, and Floer's $epsilon$ space sits in $C^infty$).

To show that this subspace is dense in $L^2$, it suffices to show that one can approximate
indicator functions.
For this, he needs the $epsilon_k$ to go to zero very fast. His explicit
construction in Lemma 5.1 (of the "unregularized gradient flow" paper) is to take a fixed cut-off function $beta$,
and approximate the characteristic function of a rectangle. The
approximation to a characteristic function is going to be a product of
terms that behave like $beta(x/delta)$ (with a better approximation
as $delta rightarrow 0$). Thus, the behaviour of the
$epsilon$-norm is going to be roughly:
[
sum_{k=0}^infty epsilon_k delta^{-k} a_k,
]
where $a_k = sup | D^k beta |$. We need this to converge for each $delta > 0$.
Floer takes $epsilon_k = (a_k k^k)^{-1}$. In particular, then, these constants are going to
$0$. Following this argument, it seems we can take the $epsilon_k$ to be on the order of $1/k!$.

Note that if the sequence $epsilon_k$ is not summable, we expect the
space to be very small. In particular, consider this norm on a
compact interval, say $[-pi, pi]$. Then, cos(x) is not in the
space.

The Floer $epsilon$ space forms a Banach algebra if $epsilon_k$ decays faster than $1/k!$.
Then,
[
sum epsilon_k |D^{(k)}(fg)|
le sum_{k=0}^infty sum_{l=0}^{k}
epsilon_k | D^{(l)} f|
|D^{(k-l)} g | binom{k}{l}
= sum_{l=0}^infty |D^{(l)} f| sum_{p=0}^{infty} binom{l+p}{p} epsilon_{p+l} | D^{(p)}g|
]
When $epsilon_{p+l} binom{l+p}{p}
le epsilon_p epsilon_l$, we are then in business.
In particular, this works for Floer's original construction.