Metric entropies of sets in abstract Wiener space

Abstract

Let ( 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 inf ɛ → 0 ɛ 2 E ( ɛ , K ) . In particular, if lim inf ɛ → 0 ɛ 2 E ( ɛ , K ) = 0 , then K is a slim set. Moreover, if K is compact and contained in the closure of B 0 H ( R ) in X, where B 0 H ( R ) : = { h ∈ H : ‖ h ‖ H < R } is a ball in H, then lim sup ɛ → 0 ɛ 2 E ( ɛ , K ) < ∞ .