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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?

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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.

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