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.
