# Estimate the metric entropy of unit ball in $L^2$ space

Let me clarify the setting I'm thinking.

For any totally bounded metric space $$(Y,d_Y)$$ and $$\varepsilon>0$$, the $$\textit{metric entropy}$$ $$N_M(\varepsilon,Y)$$ is the smallest number of closed ball $$B(y_i,\varepsilon):=\{y\in Y:d_Y(y,y_i)\leq\varepsilon\}$$, $$i=1,\dots,n$$, covering whole $$Y$$. In other words, $$N_M(\varepsilon,Y)$$ is the cardinality of the smallest $$\varepsilon$$-net of $$Y$$.

Now, consider a compact metric space $$(X,d_X)$$ and a finite borel measure $$\mu_X$$ on it. Consider the unit closed ball $$B(0,1)\subset L^2(\mu_X)$$. My questions are the following:

Q1: What is the growth rate of $$N_M(\varepsilon, B(0,1))$$ as $$\varepsilon$$ goes to $$0$$?

Q2: Do we have $$\int_0^1 \log N_M(\varepsilon, B(0,1))\,d\varepsilon<\infty$$ in general? if not, then under what kinds of condition for $$(X,d_X,\mu_X)$$ can we guarantee the finiteness?

I'm not at all familiar with these topics about the metric entropy of Hilbert space. I tried some googling but it was not successful. But it feels like natural question. So, are there any references answer my above questions? Or, are there any textbooks/papers dealing with some related contents?

A ball $$B$$ of radius $$r$$ in the Hilbert space can not be covered by finitely many balls of radius less than $$r$$. Indeed, if it can, than iterating this covering we cover $$B$$ by finitely many balls of arbitrarily small radius. This would imply that $$B$$ is precompact (totally bounded), but this implies that the space is finite-dimensional.

• note that the 𝐿2(𝜇) will be finite dimensional if the measure is atomic, and this is an "iff" with the appropriate definition of "atomic"
– am70
Jun 10, 2022 at 9:57