Closed Graph Theorem Research Articles

On closed graph theorems in topological spaces and groups

On closed graph theorems in topological spaces and groups

On Levine′s Decomposition of Continuity

AbstractA strong version of Levine′s decomposition of continuity leads to the result that a closed graph weakly continuous function into a rim-compact space is continuous. This result implies a closed graph theorem: every almost continuous closed graph function into a strongly locally compact space is continuous. An open problem of Shwu-Yeng T. Lin and Y.-F. Lin asks if every almost continuous closed graph function from a Baire space to a second countable space is necessarily continuous. This question is answered in the negative by an example.

On the closed graph theorem in topological spaces

In this paper we introduce the class of C-Suslin topological spaces and we study some properties of this class in relation to the closed graph theorem.

The range space in a closed graph theorem

The range space in a closed graph theorem

An algebraic closed graph theorem

An algebraic closed graph theorem

A note on the closed graph theorem

A note on the closed graph theorem

A variant of separability in dual systems

In (12) we introduced the concept of essential separability and used it to define two classes of locally convex spaces, δ-barrelled spaces and infra-δ-spaces, which serve as domain and range spaces respectively in certain closed graph theorems (12, Theorems 3 and 7). In this note we continue the study of these ideas. The relevant definitions are reproduced below.

Closed graph theorems for generalized inductive limit topologies

SummaryThe object of this and a subsequent paper is to investigate the locally convex structure of several strict topologies that are generalizations of R. C. Buck's strict topology β on C(S), S locally compact Hausdorff. If the topology τ of a locally convex space (lcs) (X, τ) is any of these strict topologies, then it is localizable on every absorbing disc T in X, i.e. it is the finest locally convex topology on X agreeing with τ on T. Topologies of this kind are said to be (L)-topologies. As our main tools for the analysis of the structure of strict topologies, we deduce in this paper several closed graph theorems for spaces of type (L). In particular, it is shown that every semi-Montel lcs with a fundamental sequence of bounded sets and every Bτ-complete Schwartz space belongs to the class Bτ(L) of all lcs Y with the property that every closed linear map from any (L)-space X into Y is continuous. Further closed graph theorems are established and many of the known closed graph theorems are deduced as special cases of our results. Moreover, the problem of Bτ-completeness of locally convex spaces belonging to Bτ(L) is considered.

The closed graph theorem

The closed graph theorem

A categorical approach to the closed graph theorem

A categorical approach to the closed graph theorem

On the closed graph theorem

Our main purpose is to describe those separated locally convex spaces which can serve as domain spaces for a closed graph theorem in which the range space is an arbitrary Banach space of (linear) dimension at most c, the cardinal number of the real line R. These are the δ-barrelled spaces which are considered in §4. Many of the standard elementary Banach spaces, including in particular all separable ones, have dimension at most c. Also it is known that an infinite dimensional Banach space has dimension at least c (see e.g. [8]). Thus if we classify Banach spaces by dimension we are dealing, in a natural sense, with the first class which contains infinite dimensional spaces.

The continuity of semiadditive functionals

In this paper we examine a general principle of continuity of semiadditive functionals on a metric linear space (MLS) that extends various results obtained previously in this respect. In particular, from our fundamental Theorem 1 it is easy to obtain Banach's closed graph theorem, the Banach-Steinhaus theorem, the weak base theorem, and others.

Nondiscrete mathematical induction and iterative existence proofs

The author proves a simple general theorem about complete metric spaces which forms an abstract basis of existence theorem in functional analysis and numerical analysis. He shows that this theorem, the so called induction theorem, contains the classical fixed point theorem for contractive mappings as well as the closed graph theorem. He then explains the principles of application of the induction theorem, the method of nondiscrete mathematical induction which consists in reducing the given problem to a system of functional inequalities, to be satisfied by a certain function, called the rate of convergence. The fact that the rate of convergence is defined as a function and not a number makes it possible to obtain sharp estimates valid for the whole iterative process, not only asymptotically. The method of nondiscrete mathematical induction is then illustrated by means of the example of eigenvalues of almost decomposable matrices.

The Open Mapping and Closed Graph Theorem for Embeddable Topological Semigroups

Some extensions of the open mapping and closed graph theorem are proved for certain classes of commutative topological semigroups, namely those embeddable as open subsets of topological groups. Preliminary results of independent interest include investigations of properties which “lift” from embeddable semigroups to the groups in which they are embedded, and from semigroup homomorphisms to homomorphisms of the groups.

A Closed Graph Theorem

A closed graph theorem is proved, implying that a Hausdorff locally convex space $E$ need not be barrelled if every closed linear map from $E$ into $F$ is continuous, where $F$ is a reflexive Fréchet or $LF$-space or a space of distributions.

On Komura's Closed-Graph Theorem

On Komura's Closed-Graph Theorem

Some Remarks on the Abstract Cauchy Problem

Some Remarks on the Abstract Cauchy Problem

A closed graph theorem

A closed graph theorem is proved, implying that a Hausdorff locally convex space E E need not be barrelled if every closed linear map from E E into F F is continuous, where F F is a reflexive Fréchet or L F LF -space or a space of distributions.

On Kōmura’s closed-graph theorem

Let ( α ) (\alpha ) be a property of separated locally convex spaces. Call a locally convex space E [ J ] E[\mathcal {J}] an ( α ¯ ) (\bar \alpha ) -space if J \mathcal {J} is the final topology defined by { u i : E i [ J i ] → E } i ∈ I {\{ {u_i}:{E_i}[{\mathcal {J}_i}] \to E\} _{i \in I}} , where each E i [ J i ] {E_i}[{\mathcal {J}_i}] is an ( α ) (\alpha ) -space. Then, for each locally convex space E [ J ] E[\mathcal {J}] , there is a weakest ( α ¯ ) (\bar \alpha ) -topology on E stronger that J \mathcal {J} , denoted J α ¯ {\mathcal {J}^{\bar \alpha }} . Kōmura’s closed-graph theorem states that the following statements about a locally convex space E [ J ] E[\mathcal {J}] are equivalent: (1) For every ( α ) (\alpha ) -space F and every closed linear map u : F → E [ J ] u: F \to E[\mathcal {J}] , u is continuous. (2) For every separated locally convex topology J 0 {\mathcal {J}_0} on E, weaker than J \mathcal {J} , we have J ⊂ J 0 α ¯ \mathcal {J} \subset \mathcal {J}_0^{\bar \alpha } . Much of this paper is devoted to amplifying Kōmura’s theorem in special cases, some well-known, others not. An entire class of special cases, generalizing Adasch’s theory of infra-(s) spaces, is established by considering a certain class of functors, defined on the category of locally convex spaces, each functor yielding various notions of “completeness” in the dual space.

Mackey convergence and the closed graph theorem

Mackey convergence and the closed graph theorem