Metric entropies of sets in abstract Wiener space

AbstractLet (X,H,μ) be an abstract Wiener space, E(ɛ,K) denote the metric entropy of a set K⊂X. If K is not a slim set, then we prove that 0<lim infɛ→0ɛ2E(ɛ,K). In particular, if lim infɛ→0ɛ2E(ɛ,K)=0, then K is a slim set. Moreover, if K is compact and contained in the closure of B0H(R) in X, where B0H(R):={h∈H:‖h‖H<R} is a ball in H, then lim supɛ→0ɛ2E(ɛ,K)<∞