Block sequences in nuclear Fr�chet spaces with basis

Many questions in functional analysis lead to a consideration of inductive limits of Frechet or even Banach spaces. This chapter utilizes many facts and ideas from the theory of bases, nuclear spaces and the general theory of locally convex spaces. It considers F to be the dual of a nuclear Frechet space E. F will be equipped with the strong topology. The chapter also utilizes a method which consists of first embedding E' in a space which has a basis, and then constructing a certain block basic sequence which can be approximated by a sequence in E', and an old perturbation theorem of L.

It is well known that every module M over the algebra ℒ(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ≅ = E ⊗ X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the stereotype algebra ℒ (X) of operators on X there exists a unique (up to isomorphism) stereotype space E such that M lies between two natural stereotype tensor products of E by X, $$E \circledast X \subseteq M \subseteq E \odot X.$$ . As a corollary, we show that if X is a nuclear Frechet space with a basis, then each Frechet module M over the stereotype operator algebra ℒ(X) can be uniquely represented as the projective tensor product of X by some Frechet space E, \(M = E \widehat \otimes X\).

It is proved that any infinite-dimensional non-archimedean Frechet space with a symmetric basis is isomorphic to c 0 or ?ℕ. A similar result is shown for homogeneous bases. It is also proved that any infinite-dimensional nuclear non-archimedean Frechet space with a subsymmetric basis is isomorphic to ?ℕ. In fact, much stronger results are obtained.

We establish the existence of a stochastic integral in a nuclear space setting as follows. Let , , and be nuclear spaces which satisfy the following conditions: the spaces are reflexive, complete, bornological spaces such that their strong duals also satisfy these conditions. Assume that there is a continuous bilinear mapping of into . If is an integrable, -valued predictable process and is an -valued square integrable martingale, then there exists a -valued process called the stochastic integral. The Lebesgue space of these integrable processes is studied and convergence theorems are given. Extensions to general locally convex spaces are presented.

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

Let E′ be the dual of a nuclear Frechet space E (of which the Schwartz space ℒ(ℝd) of tempered distributions is one). We give a simple sufficient condition for the strong continuity of a weakly continuous E′-valued stochastic process and as applications examine the Kolmogorov-Hahn-Delporte criteria.

This paper presents the large deviation principle for stochastic evolution equation driven by gaussian martingales taking values in duals of nuclear Frechet spaces

Mityagin posed the problem, whether complemented subspaces of nuclear infinite type power series spaces have a basis. A related more general question was asked by Pełczyński. It is well known for a complemented subspace $E$ of a nuclear infinite type power series space, that its diametral dimension can be represented by $\Delta E = \Delta {\Lambda _\infty }(\alpha )$ for a suitable sequence $\alpha$ with ${\alpha _j} \geq \ln (j + 1)$. In this article we prove the existence of a basis for $E$ in case that ${\alpha _j} \geq j$ and $\sup \tfrac {{{\alpha _{2j}}}}{{{\alpha _j}}} < \infty$.

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

One of the most fundamental properties of the power series space of analytic or entire functions is that, they are simple, i.e. bounded subsets of each of them is contained in the normal hull of a single point of the space.Infact, every nuclear Kothe space is simple and in the case of a Frechet space this property is synonymous with nuclearity ([6], p. 330).Simple character plays a key role in the study of theory of bases in Frechet spaces ([4]).Thanks to Jacob ([1]) who laid the foundation for the study of matrix transformations on simple perfect sequence spaces.The present article is a contribution in that direction.Objective of this paper is to characterize matrix transformations on nuclear Kothe-spaces and to reveal their structural behaviour.This aspects of the study of matrix transformations paves the way for a new direction which relates the structure of transformations to the associated nuclearities of the underlying spaces with bases (cf.Theorem 3.1).The last result illuminates a new direction of study, namely, the study of matrix transformation on sequence spaces, where the traditional normal topology is replaced by the fairly generalized cr/z-topology.§1.BACKGROUND AND TERMINOLOGY We expect the reader to be familiar with the rudiments of locally convex spaces and for that reason it suffices to refer to some standard text, e.g.([2], [5]).As our interest in this paper is related to sequence spaces, for the sake of completeness we recall only a very few relevant terms and results.For various other unexplained terms and facts we assume that one would turn to ([9], [11]).We write A x , as usual for the Kothe dual of a sequence space A. Unless stated otherwise, it will be taken into account that each A is equipped with the natural

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

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

In this article we discuss the relationship between domains of existence domains of holomorphy, holomorphically convex domains, pseudo convex domains, in the context of locally convex topological vector spaces. By using the method of Hirschowitz for Π n=1 ∞ C and the method used for Banach spaces with a basis we prove generalisations of the Cartan-Thullen-Oka-Norguet-Bremmerman theorem for finitely polynomially convex domains in a variety of locally convex spaces which include the following:1) N-projective limits of Frechet spaces with a basis;2) countable direct sums of Frechet spaces with a basis;3) nuclear spaces.

In this paper, it is shown that the asymptotically normable spaces are the smallest class of Frechet spaces which contains the nuclear Kothe spaces with continuous norm, the Banach spaces and is closed under e-tensor products and sub-spaces. Again our main aim will be to construct an example of a Kothe space which is Montel, admits a continuous norm, but still is not asymptotically normable. Keywords: Asymptotically normable; Frechet space; Kothe space DOI: 10.3126/bibechana.v7i0.4043BIBECHANA 7 (2011) 39-43

&#x0D; &#x0D; &#x0D; The aim of this paper is to give a notion of uniform tightness for transition probabilities on topological spaces, which assures the uniform tightness of compound probability measures. Then the upper semicontinuity of set-valued mappings are used in essence. As an important example, the uniform tightness for Gaussian transition probabilities on the strong dual of a nuclear real Frechet space is studied. It is also shown that some of our results contain well-known results concerning the uniform tightness and the weak convergence of probability measures. &#x0D; &#x0D; &#x0D;