Some criteria for nuclearity

Abstract

For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.

$ \lambda(P)$-nuclearity of locally convex spaces having generalized bases

“…However, the converse remains untrue (cf. (22], [12]). For the details of various types of bases and their applications related aspects we turn to [1], [2], [12] and [14].…”

“…This establishes that, in general λ(P ; N)-nuclearity is a weaker property than λ(P )-nuclearity. The facts and results with respect to λ(P ; N)-nuclearity are to be found in [5], [6] and [22] while for the stronger notion λ(P )-nuclearity we turn to [6], [15] and [23].…”

“…Throughout P = {(a k i )k ≥ 1} will be taken as a stable nuclear power set of infinite type (cf. [7], [22]).…”

“…But semi-reflexive DF -spaces are complete. So in view of Proposition 2.5 [22] E can be topologically identified with the Köthe space λ(Q), where…”

“…Since λ(P ) is λ(P ; N)-nuclear (cf. [22]) it follows that λ(Q) is λ(P, N)-nuclear by Corollary 2.7 [22]. However, Nelimarkka [15] informs that λ(P ; N)-nuclear DF -spaces are λ(P )-nuclear.…”