’Restricted’ Closed Graph Theorems

Abstract

The paper deals with ’restricted’ closed’ graph theorems in the following sense: Let a suitable class $\mathcal{E}$ of Mackey spaces as domain spaces, a class $\mathcal{F}$ of locally convex spaces as range spaces and for each $F\in \mathcal{F}$ a suitable class ${\mathcal{C}}_{F,\mathfrak{B}}$ of linear maps $T:E\to F(E\in \mathcal{E})$ with closed graph defined by a (general) property $\mathfrak{B}$ be given. In demand is a sufficient or even a sufficient as well as necessary condition to $F\in \mathcal{F}$ such that each $T\in {\mathcal{C}}_{F,\mathfrak{B}}$ is continuous. Note that in the situation of the well-known closed graph theorems of Pták and Kalton, $\mathcal{E}$ is the class of barrelled spaces and Mackey spaces with sequentially complete weak dual space, respectively, $\mathcal{F}$ is the class of all locally convex spaces and for each $F\in \mathcal{F}$ the class ${\mathcal{C}}_{F,\mathfrak{B}}$ is defined to be the set of all linear maps $T:E\to F(E\in \mathcal{E})$ with closed graph; a sufficient condition for $F$ in the asked sense is $B_r$-completeness (Pták [10]) and to be a separable $B_r$-complete space (Kalton [8]), respectively; a sufficient as well as necessary condition for $F$ is $A$-completeness (Zhu and Zhao [20]) and to be an $L_r$-space (Qiu [11], see [12] too), respectively. Based on the ideas developed in [4] and [2] in the present paper the notions of $\Theta$-completeness and of $(\Theta ,F,\mathfrak{B})$-spaces are introduced and a very general ’restricted’ closed graph theorem of the described type (see Theorem 2.4) is proved. Furthermore, it is shown that this main result contains both new interesting special cases and a series of known closed graph theorems, for example those due to Pták and Kalton and more generally that of Zhu and Zhao as well as that of Qiu. In addition, the paper deals with further aspects which are related to $(\Theta ,\mathcal{F},\mathfrak{B})$-spaces and motivated by the $\phi$-topologies introduced by Ruckle [13]. At last, the general considerations serve to extend the well-known inclusion theorems of Bennett and Kalton [1, Theorem 5 and 4] which establish a connection between functional analysis (weak sequential completeness and barrelledness) and summability.