Metric entropy ($H(\epsilon ,E)=\log N(\epsilon ,E)$)

For a subset $E$ of a metric space, its metric entropy at resolution $\epsilon$ is the logarithm of the least number of radius-$\epsilon$ balls covering $E$, with value infinity when no finite cover exists. It quantifies how many choices are needed to approximate an element at that resolution. If points are pairwise more than $2\epsilon$ apart, each covering ball contains at most one, giving a lower bound by their logarithmic cardinality. The logarithm base changes only a constant factor. This approximation entropy differs from Kolmogorov-Sinai entropy of a dynamical system.

Suppose outer functions in a family $B$ obey $|f(x)-f(y)|\le L{\max }_j|x_j-y_j|$, and compose them with $m$ inner functions drawn from $E$. Approximate the outer function within $\epsilon /2$ and each inner function within $\epsilon /(2L)$. The triangle inequality bounds the composed error by $\epsilon$. Multiplying the numbers of possible net choices and taking logarithms gives the stated inequality. A uniform bound on first partial derivatives supplies $L=m$ for the $C^p$ unit ball, so the exponent of a common entropy bound for the outer and inner classes is retained under function composition.

For the unit ball of $C^p({\mathbb{T}}^n)$ measured in the uniform metric, $c{\epsilon }^{-n/p}\le H(\epsilon )\le C{\epsilon }^{-n/p}\log (1/\epsilon )$. The lower bound uses disjoint bump functions of spatial scale $h$, height proportional to $h^p$ and independently chosen zero/one coefficients: there are $\exp (c{h}^{-n})$ functions separated at scale $h^p$. For the upper bound, Multivariable Jackson approximation gives a trigonometric polynomial of degree $O({\epsilon }^{-1/p})$. Its $O({\epsilon }^{-n/p})$ coefficients are bounded and can be rounded with coefficient accuracy proportional to $\epsilon$ divided by the coefficient count. Counting those rounded coefficient arrays supplies the logarithmic factor. The continuous-function unit ball without derivative bounds has infinite entropy for $\epsilon <1/2$, by an infinite sequence of disjoint height-one bumps.