TY - JOUR
T1 - Metric entropies of sets in abstract Wiener space
AU - Zhang, Xicheng
PY - 2005/8
Y1 - 2005/8
N2 - Let (X, H μ) be an abstract Wiener space, E (ε, K) denote the metric entropy of a set K ⊂ X. If K is not a slim set, then we prove that 0 < lim inf ε→0 ε2 E(ε). In particular, if lim inf3→0 ε2E (ε, 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 := {h ∈ H: ∥h∥H < R} is a ball in H, then lim supε→0 ε2 E(ε, K) < ∞.
AB - Let (X, H μ) be an abstract Wiener space, E (ε, K) denote the metric entropy of a set K ⊂ X. If K is not a slim set, then we prove that 0 < lim inf ε→0 ε2 E(ε). In particular, if lim inf3→0 ε2E (ε, 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 := {h ∈ H: ∥h∥H < R} is a ball in H, then lim supε→0 ε2 E(ε, K) < ∞.
KW - Capacity
KW - Metric entropy
KW - Slim set
KW - Small ball
UR - http://www.scopus.com/inward/record.url?scp=22944447777&partnerID=8YFLogxK
U2 - 10.1016/j.bulsci.2005.02.002
DO - 10.1016/j.bulsci.2005.02.002
M3 - Article
AN - SCOPUS:22944447777
SN - 0007-4497
VL - 129
SP - 559
EP - 566
JO - Bulletin des Sciences Mathematiques
JF - Bulletin des Sciences Mathematiques
IS - 7
ER -