STUDIA MATHEMATICA 150 (1) (2002) The non-pluripolarity of compact sets in complex spaces and the property (LB ∞ ) for the space of germs of holomorphic functions by Le Mau Hai and Tang Van Long (Hanoi) Abstract. The aim of this paper is to establish the equivalence between the non- pluripolarity of a compact set in a complex space and the property (LB ∞ ) for the dual space of the space of germs of holomorphic functions on that compact set. 1. Introduction. Let E be a Fr´ echet space with the topology defined by an increasing system {‖ · ‖ k } k≥1 of seminorms. For each k ≥ 1 put ‖u‖ ∗ k = sup{|u(x)| : ‖x‖ k ≤ 1} where u ∈ E ∗ , the topological dual space of E. We say that E has the property (  Ω) if ∀p ∃q,d> 0 ∀k ∃C> 0: ‖u‖ ∗1+d q ≤ C ‖u‖ ∗ k ‖u‖ ∗d p , ∀u ∈ E ∗ , and has the property (LB ∞ ) if ∀{ n }↑ + ∞∀p ∃q ∀n 0 ∃N 0 ,C> 0 ∀u ∈ E ∗ ∃n 0 ≤ k ≤ N 0 : ‖u‖ ∗1+ k q ≤ C ‖u‖ ∗ k ‖u‖ ∗ k p . We then write E ∈ (  Ω) (resp. E ∈ (LB ∞ )). The above properties and many others were introduced and investigated by Vogt (for example, see [11], [14]). One of the first problems raised here is to find conditions under which a Fr´ echet space has the property (LB ∞ ) or (  Ω). In [5], S. Dineen, R. Meise and D. Vogt have shown that a nuclear Fr´ echet space E has the property (  Ω) if and only if E contains a bounded subset which is not uniformly polar. In [11] they have obtained a holomorphic characterization of nuclear Fr´ echet spaces E with (  Ω) which is related to holomorphic extendability. Another problem considered here is the following. In [14] Vogt proved that the property (  Ω) implies the property (LB ∞ ). Also in [14] by an example in the space Λ(B) of K¨ othe sequences Vogt showed 2000 Mathematics Subject Classification : 32U05, 46A04, 46A63. [1]