Montel subspaces of Fréchet spaces of Moscatelli type

Abstract

In this note we show that every complemented Montel subspace F of a Fre'chet space E of Moscatelli type is isomorphic to <o or is finite-dimensional; the last case always occurs when E has a continuous norm. To do this, we first study the topology induced by E on its Montel subspaces, extending a result on Fr6chet-Montel spaces of Moscatelli type in [4].We recall that the Frechet spaces of Moscatelli type were introduced and studied by J. Bonet and S. Dierolf in [4]; the general idea behind the construction of such spaces was due to V. B. Moscatelli [7].The paper has three sections. The first one is devoted to the necessary definitions and preliminaries; in the second we prove our main result and in the third we apply it to some concrete function spaces of Frechet-Sobolev type.

The Canonical Spectral Measure in Köthe Echelon Spaces

“…Let {E j } r j=1 be any finite partition of N. Since q (1) n (P (E)x) = i∈E a n (i)|x i |, for every E ⊆ N and n ∈ N, we conclude that…”

“…Given J 2 and m(1), we apply Proposition 2.1(iii) to find m(2) > m (1) such that inf i∈J2 a m(1) (i)/a m(2) (i) = 0 and then choose a subsequence J 3 = (i 2 (s)) s∈N of J 2 with i 1 (1) < i 2 (1) and lim s→∞ a m(1) (i 2 (s)) a m(2) (i 2 (s)) = 0..…”

“…The finite variation of P in L b (λ 1 (A)) implies the existence of a positive constant M (z, n), for each n ∈ N, such that k j=1 q (1) n,Bz (P ({j})) ≤ M (z, n), k∈ N.…”

“…This is more restrictive than usual; typically only (1) and (2) are assumed. It follows from (1) -(4) and the results in Section 3 of [15] that the set function…”

Isomorphic classification of the spaces $H^k_{\rm loc}(\Bbb R)\cap L^2(\Bbb R)$

“…In this paper we study the spaces H k loc R L 2 R and represent them as Köthe spaces of Moscatelli type; the results of [7] and [2] then apply and clarify their topological structure. In order to study their isomorphic classification, we use the invariants introduced in [11] and [12] (see also [16]) and extend some results of [10] to the present situation.…”