A simpler description of the κ‐topologies on the spaces DLp,Lp,M1

Abstract

For the spaces DLp, Lp and M1, we consider the topology of uniform convergence on absolutely convex compact subsets of their (pre‐)dual space. Following the notation of J. Horváth's book we call these topologies κ‐topologies. They are given by a neighbourhood basis consisting of polars of absolutely convex and compact subsets of their (pre‐)dual spaces. In many cases it is more convenient to work with a description of the topology by means of a family of semi‐norms defined by multiplication and/or convolution with functions and by classical norms. We give such families of semi‐norms generating the κ‐topologies on the above spaces of functions and measures defined by integrability properties. In addition, we present a sequence‐space representation of the spaces DLp equipped with the κ‐topology, which complements a result of J. Bonet and M. Maestre. As a byproduct, we give a characterisation of the compact subsets of the spaces DLp′, Lp and M1.

Projective descriptions of spaces of functions and distributions

“…In [4, Proposition 6.4], a characterization of relatively compact subsets C of $$\mathcal {M}^1$$ is given: $$\exists (g,h) \in \mathcal {C}_0 \times L^1: C \subset g(h * B_{1,1})$$, $$B_{1,1} = \lbrace \mu \in \mathcal {M}^1: \left\Vert \mu \right\Vert _1 \le 1 \rbrace$$. This is not correct: A counterexample is $$C = \lbrace \delta \rbrace$$.…”

“…In Section 7, we correct an error in [4,Proposition 6.4]. We characterize weakly relatively compact subsets of M 1 and we prove a generalization of Buck's topology by a projective description of the spaces (B 𝑚 , 𝜏(B 𝑚 ,…”

“…In Section 7, we correct an error in [4, Proposition 6.4]. We characterize weakly relatively compact subsets of $$\mathcal {M}^1$$ and we prove a generalization of Buck's topology by a projective description of the spaces $$({\mathcal {B}}^m, \tau ({\mathcal {B}}^m, \mathcal {D}^{\prime m}_{L^1}))$$ and $$(\mathcal {O}_C^m, \tau (\mathcal {O}_C^m, \mathcal {O}^{\prime m}_C))$$, $$m \in \mathbb {N}_0$$.…”

Projective descriptions of spaces of functions and distributions

“…In Section 7 we correct an error in [4,Proposition 6.4]. We characterize weakly relatively compact subsets of M 1 and we prove a generalization of Buck's topology by a projective description of the spaces (B m , τ…”

Sequence space representations for translation-modulation invariant function and distribution spaces

“…The representation of function and distribution spaces by sequence spaces is a central topic in functional analysis that goes back to the pioneering work of Valdivia and Vogt [27,28] and is closely connected to the isomorphic classification of such spaces. In particular, there has been a considerable interest in the sequence space representations of the classical Schwartz spaces D L p , B, Ḃ and D ′ L p , B ′ , Ḃ′ [3,4,5,13,23,26,28]. The main goal of this article is to unify and generalize these known representations by providing sequence representations for test function and distribution spaces defined via a broad class of translation-modulation invariant Banach spaces that was recently introduced in [15].…”