Strict Inductive Limits Research Articles

Dieudonné-Schwartz theorem on bounded sets in inductive limits. II

The Dieudonné-Schwartz Theorem [1, Chapter 2, §12] has been stated for strict inductive limits. In [3] it has been extended to inductive limits. Here the result of [3] is generalized. Also, the case when each set bounded in ind lim E n {E_n} is contained, but not necessarily bounded, in some E n {E_n} is considered.

Holomorphic functions on strict inductive limits

We study strict inductive limits of Frechet Montel (FM) spaces and reflexive Frechet (RF) spaces and we obtain some interesting examples in the theory of infinite dimensional holomorphy.

Dieudonné-Schwartz theorem on bounded sets in inductive limits

The Dieudonné-Schwartz Theorem for bounded sets in strict inductive limits does not hold for general inductive limits E = ind lim E n E = {\text {ind}}\;\lim {E_n} . It does if every closed convex set in E n {E_n} is closed in E n + 1 {E_{n + 1}} . This condition is not necessary. In case all spaces E n {E_n} are normed a necessary and sufficient condition for the validity of the Dieudonné-Schwartz Theorem is given.

Barrelledness conditions onC 0 (E)

Some conditions of barrelledness are considered and studied on the spaceC 0(E), defined as follows: IfE is a real or complex Hausdorff locally convex space and $$P_E $$ is a saturated family of seminorms, defining the original topology ofE, then the vector space of all the sequences $$\bar f = \left\{ {\bar f(n): n \in \mathbb{N}} \right\}$$ inE, convergent to zero, provided with the locally convex topology $$\bar p(\bar f) = sup\left\{ {p (\bar f(n)): n \in \mathbb{N}} \right\}p \in P_E $$ is defined as the spaceC 0(E). The main result of the paper is the following characterization:C 0(E) is quasibarrelled (see [3], p. 367) if and only if,E is quasibarrelled and the strong dual ofE has property (B) (see [5], p. 30, for definition). We obtain. as a consequence, commutativity properties of the operatorC 0, acting on certain inductive limits (3.3 Theorem). We also prove thatC 0 does not commute with uncountably strict inductive limits. In particular, there are ultrabornological spacesE for whichC 0(E) is not quasibarrelled. 3.1. Example provides a completeɛ-tensor product of two complete ultrabornological spaces which is not quasibarrelled.

Bounded sets in inductive limits

The Dieudonné-Schwartz theorem for bounded sets in strict inductive limits does not hold for general inductive limits. A set B bounded in an inductive limit E = ind lim E n E = {\operatorname {ind}}\;\lim {E_n} of locally convex spaces may not be contained in any E n {E_n} . If, however, each E n {E_n} is closed in E, then B is contained in some E n {E_n} , but may not be bounded there.

On countable strict inductive limits

In this paper we prove that if E is the strict inductive limit of a sequence of Mackey spaces {En} such that for every positive integer n, the topological dual space of En, E′n, provided with the Mackey topology μ(E′n,En), is ultrabornological, then the topological dual space E′ of E, provided with the Mackey topology μ(E′,E), is ultrabornological. We also prove that if E is a strict (LF)-space and G a closed subspace of E′ [μ(E′,E)] such that E′[μ(E′,E)] /G is sequentially complete, then E′[μ(E′,E)]/G is complete.

Extending continuous linear functionals in convergence vector spaces

Let (E, r) be a convergence vector space, M a subspace of E, and p a linear functional on M continuous in the induced convergence structure. Sufflcient and sometimes necessary conditions are given that (1) (p has a continuous linear extension to the T-adherence M of M; (2) p has a continuous linear extension to E; (3) M is T-closed; (4) every T-closed convex subset of E is a(E, E')-closed. Several examples are included illustrating the extent and limitations of the theory presented. Introduction. Through introduction of an appropriate notion of local convexity, necessary and sufficient conditions are given in order that a subspace M of a convergence vector space (c.v.s.) (E, r) (H. R. Fischer, Limesrdume, Math. Ann. 137 (1959), 269-303) have the Hahn-Banach Property (H.B.P.), namely: Every continuous linear functional p on M has a continuous linear extension to E. This yields an extension of the Hahn-Banach Theorem to a class of c.v.s. satisfying a local convexity condition. Conditions are given insuring that the r-closed and weakly closed subsets of E coincide and, in a c.v.s. where this is the case, that a subspace will have the H.B.P. Prerequisite to this last result is the determination of when every continuous linear functional, p, on M has a continuous linear extension to M, the r-adherence of M. The notion of a nearly closed subspace M of (E, r) is introduced, and it is shown that for nearly closed subspaces, one can always extend p on M continuously to M and that M is r-closed. Subsequently, it is demonstrated that in a strict convergence inductive limit of Frechet spaces, M is nearly closed if and only if every such p on M extends continuously to M if and only if M is r-closed. The final section consists of examples illustrating the extent and limitations of the theory presented. In particular, we (1) provide an example of a locally convex convergence space with a closed subspace which does not have the H.B.P.; (2) provide a characterization of those subspaces M of a strict inductive limit of metrizable spaces in which every continuous linear functional on M has a continuous linear extension to M; and thus (3) characterize those subspaces of a Received by the editors May 15, 1972 and, in revised form, February 14, 1973. AMS (MOS) subject classifications (1970). Primary 46A99. Copyright X 1974, American Mathematical Society

Some results for probability measures on linear topological vector spaces with an application to Strassen's log log law

We prove some results regarding tight probability measures on real Frechet spaces and countable strict inductive limits of these spaces. These results are applied to Gaussian measures and to construct a Brownian motion on such spaces. We then prove the log log law of Strassen for this Brownian motion.

Inductive limits and partitions of unity

Let E be the inductive limit of a family of locally convex topological vector spaces El. We introduce the concept of partitions of unity in E, which extends the usual partitions of unity in function spaces. Inductive limits admitting such a partition of unity have many interesting properties, similar to those of the strict inductive limits. They also have rather unusual properties concerning duals and spaces of mappings, which seem to be unknown, even in spaces likeD(Ω) andE(Ω). Some examples of such inductive limits are quoted at the end of the paper.

ON TOPOLOGICAL VECTOR GROUPS

We study topological vector spaces over the field P of real or complex numbers, endowed with the discrete topology. These objects are called topological vector groups (for brevity, TVGs). By the conjugate E' of a locally convex TVG E we mean the set of all continuous linear mappings of E into P, where P is equipped with the usual (for the plane or the line) topology. We construct a duality theory for locally convex TVGs. In particular, we obtain an analog of the Mackey-Arens Theorem: in E there exists the strongest locally convex TVG topology compatible with the duality between E and E'. This topology is the topology of uniform convergence on all absolutely convex, weakly complete subsets of E'. Each such subset is the product of a weakly compact, absolutely convex set by a weakly complete subspace (that is, by a product of lines). In the present article we also study the connection between weakly complete subsets of a TVG and the subsets satisfying the double limit condition. The results are applied to give a proof of Eberlein's Theorem for locally convex TVGs. In addition, we prove that a subset satisfying the double limit condition in the strict inductive limit of complete, locally TVGs is necessarily contained in some limiting space. Bibliography: 8 items.

Corrections to: “On sequential convergence”

In my paper [2], Theorems 8.2 and 8.3 are false. I am indebted to Dennis Sentilles who sent me a counterexample to Theorem 8.2. The wrong statements concern LS-spaces. These are linear spaces S on which there is a metric p and a function f such that xn -x in S if and only if both p(xn, x) O-0 and f(xn) remains bounded. Without restating the further conditions imposed on p and f, we single out for the present two tractable subclasses of the class of LS-spaces: (I) LF-spaces, i.e. strict inductive limits of spaces ([1], [2, Theorem 7.7]). (II) If B is a separable Banach space, and S is its dual B* with weak-star convergence of sequences, then S is an LS-space [2, Theorem 7.1]. (Actually the same is true with Frechet in place of Banach: see [1, Theoreme 5, p. 84].) In case (I), Theorems 8.2 and 8.3 are true. The Banach-Steinhaus theorem for LF-spaces is well known [1, Theoreme 2, p. 73]. But in case (II), if B is infinitedimensional, the elements of its unit ball give pointwise bounded continuous linear forms on S which are not equicontinuous for the LS-topology. The proof of Theorem 8.2 in [2] errs in assuming that multiplication by a positive scalar is an open mapping in the relative topology of a (convex, symmetric, closed, metrizable) set. The proof of Theorem 8.3 is invalid since it rests on Theorem 8.2. Disproving the statement of 8.3 requires some further work. Here is one counterexample. Let ' be the space of continuous real functions on [0, 1] with supremum norm. Then its dual ' * is the space of finite signed measures on [0, 1]. W* with weak* sequential convergence is an LS-space. It has a countable dense set and is complete as an LS-space. The LS-topology on W* is the topology of uniform convergence on sequences {f,} in ' with f,11 ---> 0 [2, around Theorem 7.8]. By the MackeyArens theorem, the dual space of (V*, Y) is ' (see also [1, Theoreme 6, p. 85]). A sequence in W is weakly convergent if and only if it is uniformly bounded and converges pointwise. Thus the following fact contradicts Theorem 8.3:

ALMOST CLOSED LINEAR SUBSPACES OF STRICT INDUCTIVE LIMITS OF SEQUENCES OF FRÉCHET SPACES

Sufficient conditions are given for the existence, in the strict inductive limit of an increasing ing sequence of Frechet spaces, of nonclosed linear subspaces whose traces on the defining spaces are closed. These conditions are satisfied, in particular, by the Schwartz space of infinitely differentiable functions with compact support on finite-dimensional Euclidean space.

The inductive limit of uncountably many compact Abelian groups

In (2), Varopoulos examined the structure of topological groups that are the inductive limits of countably many locally compact Abelian groups. The purpose of this note is to show that the theory does not extend to the case of uncountably many groups. We give two examples, the first to show that the strict inductive limit of uncountably many compact Abelian groups need not be complete, the second to show it need not be separated by its continuous characters. The treatment of this latter half follows closely that given for topological linear spaces by Douady in (l).