Closed graph theorems and Baire spaces.
The Borel graph theorem, proved by L. Schwartz, shows that the closed graph theorem is valid for linear maps defined on and valued in most spaces encountered in analysis. Recall that a topological space is called a Polish space if it is a separable complete metrizable space and that a Souslin space is the continuous image of a Polish space.
Theorem 1 is known as the closed graph theorem. Its proof can be found in (1), (5), (7), and in many other texts in functional analysis. These proofs are based on the Baire cathegory theorem. The aim of this note is to give a simple new proof of Theorem 1 using the well-known uniform boundedness principle, which we state as Theorem.
The result that a closed linear operator mapping (all of) a Banach space into a Banach space is continuous is known as the closed-graph theorem. References (a1).
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Eulerian Graphs. Definition: A graph is considered Eulerian if the graph is both connected and has a closed trail (a walk with no repeated edges) containing all edges of the graph. Definition: An Eulerian Trail is a closed walk with no repeated edges but contains all edges of a graph and return to the start vertex.
Theorem 2.3 is probably new, but it is a simple consequence of a theorem due to A. BROWN (3). Section 4 contains necessary and sufficient conditions in order that a closed linear operator has a closed range. Some theorems in sections 2 and 4 can be generalized to the case of closed linear operators.
In this paper we prove Leray-Schauder and Furi-Pera types fixed point theorems for a class of multi-valued mappings with weakly sequentially closed graph. Our results improve and extend previous results for weakly sequentially closed maps and are very important in applications, mainly for the investigating of boundary value problems on noncompact intervals.
Find out information about closed graph theorem. If T is a linear transformation on Banach space X to Banach space Y whose domain D is closed and whose graph, that is, the set of pairs for x in D, is.
On The Closed Graph Theorem - Volume 3 Issue 1 - Alex. P. Robertson, Wendy Robertson. Skip to main content. We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
In mathematics, the closed graph theorem is a basic result which characterizes continuous functions in terms of their graphs. There are several versions of the theorem.
Some of the most important versions of the Closed Graph Theorem and of the Open Mapping Theorem are stated without proof but with the detailed reference Topics: Mathematics - Functional Analysis, Mathematics - General Topology, 46A30.
The closed graph theorem is one of the corner stones of functional analysis, both as a tool for applications and as an object for research. However, some of the spaces which arise in applications and for which one wants closed graph theorems are not of the type covered by the classical closed graph theorem of Banach or its immediate extensions.
Komura's closed-graph theorem states that the following statements about a locally convex space E (R1 are equivalent: (1) For every (a)-space F and every closed linear map u: F -4 EW(3, U is continuous. (2) For every separated locally convex topology %0 on E, weaker than X, we have J( C a Much of this paper is devoted to amplifying Komura's.
