Closed Graph Theorems and Webbed Spaces by Marc De Wilde

By Marc De Wilde

Show description

Read or Download Closed Graph Theorems and Webbed Spaces PDF

Best calculus books

A history of vector analysis : the evolution of the idea of a vectorial system

Concise and readable, this article levels from definition of vectors and dialogue of algebraic operations on vectors to the concept that of tensor and algebraic operations on tensors. It also includes a scientific research of the differential and necessary calculus of vector and tensor capabilities of area and time.

Real and Abstract Analysis: A modern treatment of the theory of functions of a real variable

This ebook is to start with designed as a textual content for the path frequently referred to as "theory of features of a true variable". This path is at the present cus­ tomarily provided as a primary or moment 12 months graduate path in usa universities, even if there are indicators that this type of research will quickly penetrate higher department undergraduate curricula.

Volume doubling measures and heat kernel estimates on self-similar sets

This paper experiences the next 3 difficulties: while does a degree on a self-similar set have the quantity doubling estate with recognize to a given distance? Is there any distance on a self-similar set less than which the contraction mappings have the prescribed values of contractions ratios? And while does a warmth kernel on a self-similar set linked to a self-similar Dirichlet shape fulfill the Li-Yau sort sub-Gaussian diagonal estimate?

Extra resources for Closed Graph Theorems and Webbed Spaces

Sample text

L. 1. l. ,x 1 )B o C 2CB. Bi C 29 m 1 (*). Take now 1 9 " = c o ( 9 u 29 ' ) ; n it is a bornivorous disk of E, such that 9" L = e. It is obvious that a bornivorous disk of E induces a bornivorous disk of L, thus we have proved that Lb has the topology induced by E 0 • This implies as above that L is bornological if E is bornological. See [110] for the ultrabornological case. Let us now turn to criteria for dual spaces to be bornological. As far as applications are concerned, the main result is the following, due to L.

Let E- E be 41 such that (x 0 ,x'> = 1 and denote by T the projection x - x - x 0 of E onto L. Assume first that x' is locally bounded. Then T maps bounded sets of E into bounded sets of L, If 9 is a bornivorous disk of L, T- 1 9 is a bornivorous disk of E and 9 = L n T- 1 e. If 9 1 is a bornivorous disk of E, 9 1 n L is bornivorous in L. Hence Lb has If E is bornological, the topology induced by Eb. L is thus bor- nological too. Assume now that x' bounded sequence x ~ X m is not locally bounded.

S. equipped with the finest locally convex to- pology. Hence, for instance, prop. 16 is still true when U is the class of barrelled spaces, ultrabornological spaces, etc. PROPOSITION Ilo3o17o [Mahowald, 7 ] . j ! 'iFis the class of all Banach spaces, led spaces. ~U ~ &u are equal to the class of all barrel- Every barrelled space belongs to ~U , hence to &U • Indeed, let E be barrelled and R be a linear relation of E into a Banach space F such that ~(R) E X = E and G(R) is closed in F. For every x E- E, if xRy, then xRy 1 ~ y 1 E- y + R(O) • Moreover, R(O) is a vector subspace of F and it is closed, since 22 {0} R(O) = G(R) n ( {0} x R) X • Thus R induces a linear map T of E into F/R(O), where F/R(O) is another Banach space, and G(T) has a closed graph, since G(T) {0} with = I ( {0} G(R) x R(O) R(O)) X ' C G(R).

Download PDF sample

Rated 4.96 of 5 – based on 15 votes