Chet Spaces Research Articles

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.

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].

Dominating sequences and functional equations

The notion of domination of sequences has been used by a number of mathematicians in the past with slight variations of terminology. Indeed, the domination between pair of sequences was considered by Banach ([4], p. 1638; [5], p. 11) around the year 1925. Later, Arsove ([2], [3]) used these notions in the s tudy of Schauder basis in Fr~chet spaces and constructed some examples of such sequences (for domination of infinite matrices, one is referred to [9], [10]). However, these concepts gained independent importance in the work of Singer [23] who also employed these notions to reformulate certain results on sequence spaces. He dealt with all the problems in Banach spaces. In this note, we give various equivalent conditions for strict domination in terms of functional equations in topological vector spaces, characterize a base through the notions of domination and strict domination, and apply them to obtain a result on weakly ~-unconditionally Cauchy ser~es -a concept introduced in [12] (cf. also Chapter 3 of [13]).

Discontinuous Translation Invariant Functionals

Let $G$ be an infinite $\sigma$-compact locally compact group. We shall study the existence of many discontinuous translation invariant linear functionals on a variety of translation invariant Fréchet spaces of Radon measures on $G$. These spaces include the convolution measure algebra $M(G)$, the Lebesgue spaces ${L^p}(G)$, where $1 \leq p \leq \infty$, and certain combinations thereof. Among other things, it will be shown that $C(G)$ has many discontinuous translation invariant functionals, provided that $G$ is amenable. This solves a problem of G. H. Meisters.

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

Fréchet spaces with nuclear Köthe quotients

Each separable Fréchet non-Banach space $X$ with a continuous norm is shown to have a quotient $Y$ with a continuous norm and a basis. If, in addition, $Y$ can be chosen to be nuclear, we say that $X$ has a nuclear Köthe quotient. We obtain a (slightly technical) characterization of those separable Fréchet spaces with nuclear Köthe quotients. In particular, separable reflexive Fréchet spaces which are not Banach (and thus Fréchet Montel spaces) have nuclear Köthe quotients.

Division by Holomorphic Functions and Convolution Equations in Infinite Dimension

Let $E$ be a complex complete dual nuclear locally convex space (i.e. its strong dual is nuclear), $\Omega$ a connected open set in $E$ and $\mathcal {E}(\Omega )$ the space of the ${C^\infty }$ functions on $\Omega$ (in the real sense). Then we show that any element of $\mathcal {E}’(\Omega )$ may be divided by any nonzero holomorphic function on $\Omega$ with the quotient as an element of $\mathcal {E}’(\Omega )$. This result has for standard consequence a new proof of the surjectivity of any nonzero convolution operator on the space $\operatorname {Exp} (E’)$ of entire functions of exponential type on the dual $E’$ of $E$. As an application of the above division result and of a result of ${C^\infty }$ solvability of the $\overline \partial$ equation in strong duals of nuclear Fréchet spaces we study the solutions of the homogeneous convolution equations in $\operatorname {Exp} (E’)$ in terms of the zero set of their characteristic functions.

Complex homomorphisms of the algebras of holomorphic functions on Fr�chet spaces

Complex homomorphisms of the algebras of holomorphic functions on Fr�chet spaces

Conditions for certain forms of completeness in the class of projective limits of sequences of inductive limits of sequences of Fr�chet spaces

Conditions for certain forms of completeness in the class of projective limits of sequences of inductive limits of sequences of Fr�chet spaces

Separating sets with relative interior in Fr�chet spaces

A separation theorem, valid in infinite dimensional spaces, and involving the relative interior of the sets to be separated, will be extended to Frechet spaces. This theorem will be elucidated by means of a few examples. The second separation theorem is a generalization of an existing separation theorem, valid in Frechet spaces. This paper consists of two parts: part I contains the first theorem, the second part contains the second generalization.

Block sequences in nuclear Fr�chet spaces with basis

It is shown that a block sequence in a nuclear Frechet space with a basis has a block extension if and only if the subspace it generates is complemented. In addition, a short proof is given of the following result of Dubinsky and Robinson: a nuclear Frechet space is isomorphic toΩ = RN, N = {1,2,...} if it has a basis such that any block sequence with blocks of length ≤2 of any permutation of this basis has a block extension. It is shown that a similar result holds without considering permutations of the basis if the length of the blocks is arbitrary.

Extending homeomorphisms in compactification of Fréchet spaces

In the study of extending homeomorphisms in the compactification of Fréchet spaces into the Hilbert cube $Q$, it is shown that for a given homeomorphism $f$ of $s$ onto itself and a closed subset $K$ of $s$ satisfying property $Z$ in $s$, there is a homeomorphism $g$ of $s$ onto itself such that $gf{g^{ - 1}}{|_{g(K)}}$ extends to a homeomorphism of $Q$ onto $Q$. The proof is by employing the rather useful shifting homeomorphism on $Q$.

Similar bases and isomorphisms in Fr�chet spaces

Similar bases and isomorphisms in Fr�chet spaces