Mackey space

Abstract: AwrpaAcT. A P-space is a completely regular Hausdorff space X in which every G5 is open. It is shown that the generalized strict topologies f and flo coincide on C*(X), and that strong measuretheoretic properties hold; in particular, (C*(X), f,) is always a strong Mackey space. As an application, an example is constructed of a non-quasi-complete locally convex space in which closed totally bounded sets are compact.

Abstract: Several results and examples about locally bounded sets of hol(} morphic mappings defined on certain classes of locally convex spaces (Baire spaces, (DF)-spaces, C(X)-spaces) are presented. Their relation with the classification of locally convex spaces according to holomorphic analogues of barrelled and bornological properties of the linear theory is considered. Introduction. The main topic discussed in this article is the classification of locally convex spaces according to the holomorphic analogues of barrelled, bornological and quasibarrelled properties introduced for the linear theory. This question is related to the study of locally bounded sets of holomorphic mappings defined on open subsets of locally convex spaces. A satisfactory classification can only be obtained in the case of special classes of locally convex spaces, e.g., metrizable spaces, Baire spaces, (DF)-spaces and spaces of type C(X). Some positive results are presented and several examples will show the essential differences between the linear and the holomorphic theory. We shall use standard notations of locally convex spaces as in [21 and 22] and of infinite holomorphy as in [15 and 33]. The word "space" will mean separated locally convex topological vector space. If E is a space, cs(E) denotes the set of all continuous seminorms on E. A family X of mappings defined on an open subset U of a space E with values in a normed space F is called locally bounded if for every x E U there is a neighbourhood V of x contained in U and M > 0 with llf(Y)ll < M for every f E X and y E V. Now we describe briefly the organization of this paper. The first section discusses various extensions of the classical Hartogs' theorem on separate analyticity. According to results of Arias de Reyna [3] and Valdivia [39] a product of two metrizable Baire spaces need not be Baire. Consequently Theorem 1.2 is a proper extension of [6, Corollary 6.2]. In g2 we give a complete characterization of the spaces of type C(X) which are holomorphically barrelled and holomorphically quasibarrelled (cf. [5]). Dineen has recently proved in [16] that every weakly holomorphic mapping defined on any open subset of CN x C(N) is holomorphic. This result provides the first example of a holomorphically Mackey space (cf. [5]) which is not holomorphically quasibarrelled. In §3 the result of Dineen is extended and more examples and properties of holomorphically Mackey spaces are given. In the last section we present various results, on spaces related to (DF)-spaces, concerning the following topics: sequential completeness of H(U, F) endowed with the compact open topology, local boundedness of sequences of holomorphic mappings and the equivalence of local boundedness of sets and sequences on separable spaces. Received by the editors April 14, 1986 and, in revised form, July 15, 1987. 1980 Mathematics Subject Classification (1985 Relvision). Primary 46G20; Secondary 46E10. (D1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

Abstract: Let X be a completely regular space and let E be a locally convex space. Denote by Crc(X, E) the space of all continuous E-valued functions on X with relatively compact range. The dual space of Crc(X, E) under the uniform topology is a space M(B, E') of U-valued measures (see [5]). In case E is a locally convex lattice, M(B, E') becomes a lattice. In [3] the author defined the topologies fl and fl, on C,c(X, E). The corresponding dual spaces are the spaces M~(B, E') and M~(B, E') of all z-additive and all a-additive members of M(B, E') respectively. In this paper we look into the problem of the [3(fl 0 equicontinuity of weakly compact subsets of the positive cone of M~(B, E') (M,,(B, E')) when E is a locally convex lattice. In the scalar case it is known that each such set is fl(flO equicontinuous (see [12]). It is shown here that this is not always true in the vector-valued case. More specifically it is shown that, for a completely regular space X, the following assertions are equivalent: (1) X is compact (pseudocompact). (2) For each Banach lattice E, fl(fll) is the topology of uniform convergence on the weakly compact subsets of the positive cone of Me(B, E') (M~(B, E')). In case E is finite dimensional, it is shown that (C~(X, E), ill) is a strong Mackey space and that (Crc(X, E), fl) is a strong Mackey space iff (Cb(X), fl) is a strong Mackey space [Cb(X) is the space of all bounded continuous real functions on X]. Finally it is proved that, for E a Banach space and ), = fl or//1 the space (C,c(X, E), 7) is a strong Mackey space iff it is Mackey.

Abstract: An important theorem of Kakutani and Mackey characterizes an infinite dimensional real (complex) Hilbert space as an infinite dimensional real (complex) Banach space whose lattice of closed subspaces admits an orthocomplementation. This result, also valid for quaternionic spaces, has proved useful as a justification for the unique role of Hilbert space in quantum theory. With a like application in mind, we present in the present paper a number of characterizations of real and complex Hilbert space in the class of locally convex spaces. One of these is an extension of the Kakutani-Mackey result from the infinite dimensional Banach spaces to the class of all infinite dimensional complete Mackey spaces. The implications for the foundations of quantum theory are discussed.