Series expansions in Fréchet spaces and their duals, construction of Fréchet frames

Abstract

We analyze the construction of a sequence space Θ, resp. a sequence of sequence spaces, in order to have {g i } ∞ i=1 as a Θ-frame or Banach frame for a Banach space X, resp. pre-F-frame or F-frame for a Fréchet space X F = ∩ s∈N0 X s , where {X s } s∈N0 is a sequence of Banach spaces.

Analysis of conditions for frame functions, examples with the orthogonal functions

“…We will use notation and notions as in [24]. (X, · ) is a Banach space and (X * , · * ) is its dual, (Θ, |· |) is a Banach sequence space and ( …”

“…This paper is closely connected to [24] and we refer to [24] for the background material. In order to keep the information about the sources for our investigations, we quote the same literature as in [24].…”

“…In order to keep the information about the sources for our investigations, we quote the same literature as in [24].…”

“…for X 0 ). We have investigated in [24] constructions of a CB-space Θ (resp. a sequence {Θ s } s∈N 0 of CB-spaces), so that given Θ-Bessel sequence…”

Shrinking and boundedly complete Schauder frames in Fréchet spaces

“…Unconditional atomic decompositions allowed them to prove James-type results characterizing shrinking and boundedly complete unconditional atomic decompositions in terms of the containment in the Banach space of copies of ℓ 1 and c 0 respectively. Very recently, Pilipovic and Stoeva [32] (see also [31]) studied series expansions in (countable) projective or inductive limits of Banach spaces. In this article we begin a systematic study of Schauder frames in locally convex spaces, but our main interest lies in Fréchet spaces and their duals.…”