Fr Chet Spaces Research Articles

Resolutions of the identity in Fr�chet spaces

It is a classical result that every Bade σ-complete Boolean algebra of (selfadjoint) projections in a separable Hilbert space coincides with the projections forming the resolution of the identity of some bounded selfadjoint operator. This result is extended to the setting of separable Frechet spaces. Namely, every Bade σ-complete Boolean algebra of projections in such a space coincides with the resolution of the identity of some (continuous) scalar-type spectral operator having spectrum a compact subset ofℝ.

Weakly sequentially complete Fr�chet spaces of integrable functions

For a Frechet space E let \(L^1(\mu , E)\) be the space of all E-valued integrable functions with respect to a probability measure \(\mu \) and, when E is a real Frechet space, let \(L^1 ({\cal G}, E)\) be the space of all real-valued integrable functions with respect to an E-valued vector measure \({\cal G}\). We prove that if E is weakly sequentially complete then both \(L^1(\mu , E)\) and \(L^1({\cal G}, E)\) are weakly sequentially complete. We also prove that if E is a Frechet lattice, then E is weakly sequentially complete if and only if E does not contain any (lattice) copy of c0. By combining these main theorems, we also give some results about the existence of copies of c0 in \(L^1(\mu , E)\) and \(L^1({\cal G}, E)\).

Ordinary differential equations with a continuous right-hand side in Fr�chet spaces

Ordinary differential equations with a continuous right-hand side in Fr�chet spaces

Peano's theorem fails for infinite-dimensional Fr�chet spaces

Peano's theorem fails for infinite-dimensional Fr�chet spaces

Some New Classes in Topological Sequence Spaces Related to $L_r$-Spaces and an Inclusion Theorem for K(X)-Spaces

The aim of the present paper is to get inclusion theorems for K(X)-spaces, that is, sequence spaces over any Frchet space X endowed with a K-topology (e.g. domains of operator valued matrices). Since Kalton’s closed graph theorem is an essential tool to get inclusion theorems in the case that Xequals the set of all complex numbers and since domains of operator valued matrices are not necessarily separable FK(X)-spaces we can no longer make use of FK-space theory. Therefore, it is necessary to develop new ideas to get inclusion theorems. For this we introduce two new classes of K(X)-spaces and prove a closed graph theorem for inclusion maps. One of them is closely related to the class of L_r -spaces introduced by Jinghui Qiu and to the closed graph theorem of J. Qiu, the other is connected with a well-known result of K. Zeller in summability theory. As an immediate corollary of the inclusion theorem proved in this paper we get a generalization of a theorem of Mazur-Orlicz type due to the authors.

Some results on Fr�chet spaces with the density condition

Some results on Fr�chet spaces with the density condition

The bidual of a distinguished Fr�chet space need not be distinguished

The bidual of a distinguished Fr�chet space need not be distinguished

Random sequences in Fr�chet spaces

This article deals with the generation of arbitrarily distributed sequencesΦ of random variables in a Frechet space, using sequences ofcanonical random variables (c.r.v.)-i.e., independently uniformly distributed random variables taking real values in the unit interval [0, 1)-orcanonical random digits (c.r.d.)-i.e., independently uniformly distributed random variables taking integer values in some finite interval [0,B−1]. Two main results are established. First, that the members of a sequence of real random variables in [0, 1) are c.r.v. if and only if all the digits of all thebase- B digital representations of the members of the sequence are c.r.d. Secondly, that, given any sequenceΦ of random variables in a Frechet space, there is a sequenceΨ of functionsψn(ξ1,ξ2, ...,ξn), forn=1, 2, 3,... (whereξ1,ξ2,...,ξn,... are c.r.v.) which is distributed identically toΦ.

The Ramsey theorem and complemented basic sequences in Fr�chet spaces with absolute finite-dimensional decompositions

The Ramsey theorem and complemented basic sequences in Fr�chet spaces with absolute finite-dimensional decompositions

A characterization of totally reflexive Fr�chet spaces

A characterization of totally reflexive Fr�chet spaces

On stein manifolds M for whichO(M) is isomorphic toO(?n) as Fr�chet spaces

We give a characterization of Stein manifolds M for which the space of analytic functions,O(M), is isomorphic as Frechet spaces to the space of analytic functions on a polydisc interms of the existence of a plurisubharmonic function on M with certain properties. We discuss some corollaries of this result and give some examples.

A twisted Fr�chet space with basis

In this note we show that the twisted Frechet and (LB)-spaces constructed by the second author in [6, § 1] and which were known not to have unconditional bases may, however, have a basis.

Nuclear Fr�chet spaces with basis do not have the three-space property

A proper ty (P) that locally convex spaces may enjoy is termed a three-space property if, whenever E contains a subspace ( = closed subspace) F such that F and ElF have (P), then E too has (P). In many situations it is important to know whether a given property (P) is a three-space property. Here we shall prove that the property of having a basis is not a three-space proper ty for nuclear Fr6chet spaces. This will follow from showing that the proper ty of being not twisted is not a three-space property and from the result in [2; w 2].

Strict regularity of a separable Fr�chet space that does not have nuclear K�the quotient

Strict regularity of a separable Fr�chet space that does not have nuclear K�the quotient

Stochastic basis in Fr�chet space

Stochastic basis in Fr�chet space

A Fr�chet space which has a continuous norm but whose bidual does not

A Fr�chet space which has a continuous norm but whose bidual does not

Complemented basic sequences in nuclear Fr�chet spaces with finite dimensional decompositions

Complemented basic sequences in nuclear Fr�chet spaces with finite dimensional decompositions

On bounded approximation properties of spaces of holomorphic functions on certain open subsets of strong duals of nuclear Fr�chet spaces

On bounded approximation properties of spaces of holomorphic functions on certain open subsets of strong duals of nuclear Fr�chet spaces

A result on equicontinuous sets of operators on nuclear Fr�chet spaces related to the bounded approximation property

A result on equicontinuous sets of operators on nuclear Fr�chet spaces related to the bounded approximation property

Every nuclear Fr�chet space is a quotient of a K�the Schwartz space

Every nuclear Fr�chet space is a quotient of a K�the Schwartz space