Klaus D. Bierstedt

Biduals of weighted banach spaces of analytic functions

For a positive continuous weight function ν on an open subset G of C N , let Hv ( G ) and Hv 0 ( G ) denote the Banach spaces (under the weighted supremum norm) of all holomorphic functions f on G such that ν f is bounded and ν f vanishes at infinity, respectively. We address the biduality problem as to when Hν ( G ) is naturally isometrically isomorphic to Hν 0 ( G )**, and show in particular that this is the case whenever the closed unit ball in Hν 0 ( G ) in compact-open dense in the closed unit ball of Hν ( G ).

Köthe Sets and Köthe Sequence Spaces

Publisher Summary This chapter discusses the Kothe sets and Kothe sequence spaces. Echelon and co-echelon spaces had been studied by G. Kothe (and O. Toeplitz) prior to the development of general tools available through the present day theory of topological vector spaces; Kothes early work with sequence spaces has helped point the way in establishing a general theory. In its turn, however, this general theory has been successfully utilized in the study of sequence spaces, while echelon and co-echelon spaces have continued to serve as a ready source for examples and counter examples. The chapter presents the fundamental definitions and establish the notation and treats the role of the space K p in the duality of echelon and co-echelon spaces. A condition on the Kothe matrix A is considered and is preferred to phrase in terms of the corresponding decreasing sequence—namely, the sequence space analog of the property that is called “regularly decreasing.”

Dual Density Conditions in (DF)— spaces, I

We define the two “dual density conditions” (DDC) and (SDDC) for locally convex topological vector spaces and study them in the setting of the class of (DF)- spaces (originally introduced by A. Grothendieck [14]). We show that for a (DF)- space E, (DDC) is equivalent to the metrizability of the bounded subsets of E, and prove that such a space E has (DDC) resp. (SDDC) if and only if the space l∞(E) of all bounded sequences in E is quasibarrelled resp. bornological.As a consequence, we can then characterize the barrelled spaces % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

Distinguished Echelon Spaces and the Projective Description of Weighted Inductive Limits of Type (X)

{\cal L}_b(\lambda_1,\ E)

Weighted (LF)‐spaces of Continuous Functions

of continuous linear mappings from a Köthe echelon space λ1into a locally complete (DF)- space E; for purposes of a comparison, we also provide the corresponding characterization of the quasibarrelled resp. bornological (DF)- tensor products (λ1)b′ ⊗εE. Our results on the (DF)- spaces of type% MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

Biduality in Fréchet and (LB)-spaces

{\cal L}_b(\lambda_1,\ E)

Projective Descriptions of Weighted Inductive Limits: The Vector-Valued Cases

and (λ1)b′) ⊗εE are of special interest in view of the recent negative solution, due to J. Taskinen (see [25]), of Grothendieck’s “problème des topologies” ([15]). — In part II of the article, we will treat weighted inductive limits of spaces of continuous functions and their projective hulls (cf. [6]) as an application.In his study of ultrapowers of locally convex spaces, S. Heinrich [16] had found it necessary to introduce the “density condition”. Our article [2] investigated this condition, mainly in the setting of Fréchet spaces, and with applications to distinguished echelon spaces λ1. However, on the way to the main theorems of [2], it became apparent that the “right” setting for most of this material was a dual reformulation of the density condition in the context of (DF)- spaces, and this observation prompted the present research.

PROJECTIVE DESCRIPTION OF WEIGHTED (LF)-SPACES OF HOLOMORPHIC FUNCTIONS ON THE DISC

Publisher Summary This chapter discusses the distinguished echelon spaces and provides a projective description of weighted inductive limits of type ұdҞ(X). The general problem of a projective description for weighted inductive limits of spaces of continuous and holomorphic functions was raised. This problem remains open, but a new, very weak, condition sufficient for distinguishedness of echelon spaces is presented in the chapter. The property generalizes the quasi-normable and the reflexive (or, equivalently, Montel) cases of distinguished echelon spaces λ1. The chapter proves that even the weaker hypothesis is sufficient to imply that λ1 is distinguished. The chapter also demonstrates that the arguments used in the sequence space setting can be modified to yield a similar, rather general, result on the topological equality.

Some Recent Results on VC(X)

Abstract chemical structure image.

An operator representation for weighted spaces of vector valued holomorphic functions

Let E be a Frechet space (resp. an (LB)-space indnEn) and II a topological subspace of E (resp. H = indnHn for a sequence of norm subspaces Hn of En with Hn ⊂Hn+1). We give necessary and sufficient conditions that E is canonically (topologically isomorphic to) the inductive bidual (H′b)′i or (even) the strong bidual Hb″ of H. This characterization is applied to weighted spaces of holomorphic functions, weighted inductive limits of spaces of holomorphic functions, spaces of differentiable functions with Holder conditions and other examples in distribution theory, holomorphy and Kothe sequence spaces.