By V. K. Dobrev, G. Mack, V. B. Petkova, S. G. Petrova, I. T. Todorov
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SEQUENCES AND FAMILIES OF CONTINUOUS, INTEGRABLE AND HOLOMORPHIC FUNCTIONS Valdivia's research career did, however, not start with the explosion of 1968. In his doctoral dissertation [ I ] he considers concepts which are usually defined for maps from a topological space into a metric space or a uniform space. He characterizes uniformly convergent, equicontinuous, Cauchy, and quasi-uniformly convergent sequences of maps from a metric space X into a metric space Y, and then uses the characterizations to define these concepts when X let and be a topological space, Y X Y a regular topological space, and (f,) a sequence of continuous maps from X a map f : X+Y.
U,x> = 1 for all Let G There exists u E E' such XEG. 5. CLOSED GRAPH AND OPEN MAPPING THEOREMS It is remarkable that barrelled spaces, originally introduced to generalize the uniform boundedness principle, also play a role in closed graph theorems. M. (1)). e. for every neighborhood W of -1 the closure of u (W) is a neighborhood of 0 in E (19, 934, a space F 0 in F 6 . 1 V. (8)). , such that G is the dual of F equipped f with 7 . 8). e. coincides with E'. Analogously E is a f Ptak space (or B-complete) if every 7 -closed subspace of E' is o(E1,E)closed {19,§34,2).
ZOl)), then the Schwartz). F are reflexive (resp. Montel, n The mathematical works of Manuel Valdivia Baire-hyperplane spaces [68] are those in which every union of a sequence of closed hyperplanes has a void interior. Every metrizable (ZY)space is a Baire-hyperplane space. Let E be either an infinite- dimensional, separable Frechet space or a locally complete inductive limit of infinite-dimensional, separable Frechet spaces. A construction similar to one given in [771 shows that there exist two dense subspaces F of and G E such that G is ultrabornological and a hyperplane in F, and F is barrelled and not the inductive limit of Baire-hyperplane spaces.