  We develop new upper and lower bounds on the $\varepsilon$-entropy of a unit
ball in a reproducing kernel Hilbert space induced by some Mercer kernel $K$.
Our bounds are based on the behaviour of eigenvalues of a corresponding
integral operator. In our approach we exploit an ellipsoidal structure of a
unit ball in RKHS and a previous work on covering numbers of an ellipsoid in
the euclidean space obtained by Dumer, Pinsker and Prelov.
  We present a number of applications of our main bound, such as its tightness
for a practically important case of the Gaussian kernel. Further, we develop a
series of lower bounds on the $\varepsilon$-entropy that can be established
from a connection between covering numbers of a ball in RKHS and a quantization
of a Gaussian Random Field that corresponds to the kernel $K$ by the
Kosambi-Karhunen-Lo\`eve transform.
