Showing papers on "Banach space published in 1978"

Real and Functional Analysis

Abstract: This book is meant as a text for a first-year graduate course in analysis. In a sense, the subject matter covers the same topics as elementary calculus - linear algebra, differentiation, integration - but treated in a manner suitable for people who will be using it in further mathematical investigations. The book begins with point-set topology, essential for all analysis. The second part deals with the two basic spaces of analysis, Banach and Hilbert spaces. The book then turns to the subject of integration and measure. After a general introduction, it covers duality and representation theorems, some applications (such as Dirac sequences and Fourier transforms), integration and measures on locally compact spaces, the Riemann-Stjeltes integral, distributions, and integration on locally compact groups. Part four deals with differential calculus (with values in a Banach space). The next part deals with functional analysis. It includes several major spectral theorems of analysis, showing how one can extend to infinite dimensions certain results from finite-dimensional linear algebra; a discussion of compact and Fredholm operators; and spectral theorems for Hermitian operators. The final part, on global analysis, provides an introduction to differentiable manifolds. The text includes worked examples and numerous exercises, which should be viewed as an integral part of the book. The organization of the book avoids long chains of logical interdependence, so that chapters are as independent as possible. This allows a course using the book to omit material from some chapters without compromising the exposition of material from later chapters.

Some results on Banach spaces without local unconditional structure

Abstract: © Foundation Compositio Mathematica, 1978, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Periodic Solutions of Semilinear Parabolic Equations

Abstract: Publisher Summary This chapter describes the methods of nonlinear functional analysis, namely, fixed-point theorems in ordered Banach spaces, to prove existence and multiplicity result for periodic solutions of semilinear parabolic differential equations of the second order. The oldest method for the study of periodic solutions of differential equations is to find fixed points of the Poincare operator. Subsequently in the case of parabolic equations, it turns out that the Poincare operator is compact in suitable function spaces. Even in the case of the general semilinear parabolic equations, this operator is strongly increasing. Having seen that the Poincare operator is strongly increasing, it is clear that the problem can be included in the general framework of nonlinear equations in ordered Banach spaces. Hence, by applying other general fixed-point theorems for equations of this type, it is possible to obtain further existence and multiplicity results.

Some Minimax Theorems and Applications to Nonlinear Partial Differential Equations

Abstract: Publisher Summary This chapter reviews some existence theorems for critical points of a real-valued function on a real Banach space and to apply these results to elliptic and hyperbolic partial differential equations The abstract results on critical points are obtained using minimax arguments Applications to elliptic equations are thereafter provided for the same A new proof is given for a recent result of Ahmad et al , as well as some variants of their result The work on abstract results on critical points is applied to hyperbolic problems

Stability of isometries

Abstract: A map T: £ -> F(E, F Banach spaces) is called an e-isometry if | \\T(x) T(y)\\ ||x -y\\ \ < e whenever x, y G E. Hyers and Ulam raised the problem whether there exists a constant k, depending only on E, F, such that for every surjective e-isometry T: E -* F there exists an isometry /: E -> F with ||r(x) /(x)|| < ice for every x G E. It is shown that, whenever this problem has a solution for E, F, one can assume k < 5. In particular this holds true in the finite dimensional case.

On $\Phi $-Convexity in Extremal Problems

Abstract: For a class of functions $\Phi $ on an arbitrary set X, $\Phi $-convex subsets of X and functions on X are defined, the latter being least upper bounds of some functions from $\Phi $. Also the generalized Fenchel transform and $\Phi $-subgradients are determined and their properties investigated.$\Phi $-convexity and $\Phi $-subdifferentiability of lower-semicontinuous functions on metric spaces are examined with respect to special important families $\Phi $. Among related results, we present a theorem on the existence of minimizing points of nonlinear functions on Banach spaces and extensions of the notion of Holder continuity. The relevance of the theory to perturbed extremal problems is indicated.

The three space problem for locally bounded $F$-spaces

Abstract: Let X be an F-space, and let Y be a subspace of X of dimension one, with X/Y = lp (0 p oo) Provided p ~ 1, X ~lp; however if p = 1, we construct an example to show that X need not be locally convex More generally we show that Y is any closed subspace of X, then if Y is an r-Banach space (0 r:5 1) and XI Y is a p-Banach space with p r S 1 then X is a p-Banach space; if Y and XI Y are B-convex Banach spaces, then X is a B-convex Banach space We give conditions on Y and XI Y which imply that Y is complemented in X We also show that if X is the containing Banach space of a non-locally convex p-Banach space (p 1) with separating dual, then X is not B-convex COMPOSITIO MATHEMATICA, Vol 37, Fasc 3, 1978, pag 243-276 Sijthoff & Noordhoff International Publishers Alphen aan den Rijn Printed in the Netherlands

Some Critical Point Theorems and Applications to Semilinear Elliptic Partial Differential Equations.

Abstract: : Variational methods are used to obtain some new existence theorems for critical points of a real valued function on a Banach space. Applications are made to semilinear elliptic boundary value problems. (Author)

Application of Nonlinear Semigroup Theory to Certain Partial Differential Equations

Abstract: Publisher Summary This chapter highlights the facts about nonlinear semigroups in arbitrary Banach spaces and an exposition of some applications and extensions of this general theory to certain nonlinear partial differential equations of parabolic type. The chapter focuses on the applicability of the abstract considerations to several concrete problems, to explain what useful information semigroup theory provides and what it does not, and to demonstrate how in certain cases, various partial differential equations techniques can be used to extend the theoretical understanding. The chapter presents the basic working information about nonlinear semigroups, including Crandall–Liggett generation theorem and various regularity and perturbation results. It also presents the nonlinear Chernoff theorem and several related topics, with a view toward the applications. These are the porous medium and related equations, certain variational and quasi-variational inequalities of evolution, and Bellman's equation.

Nonreflexive spaces of type 2

Abstract: The nonreflexive and uniformly nonoctahedral spacesX pgr are known to be of typep if 1≦p<2 and ρ is sufficiently large. It is shown thatX ρ is of type 2 if ρ>2.

Characterisation of normed linear spaces with Mazur's intersection property

Abstract: Normed linear spaces possessing the euclidean space property that every bounded closed convex set is an intersection of closed balls, are characterised as those with dual ball having weak* denting points norm dense in the unit sphere. A characterisation of Banach spaces whose duals have a corresponding intersection property is established. The question of the density of the strongly exposed points of the ball is examined for spaces with such properties.

The duality between Asplund spaces and spaces with the Radon-Nikodym property

Abstract: A Banach spaceX is an Asplund space (a strong differentiability space) if and only ifX * has the Radon-Nikodym property.

Some Examples Concerning Rotundity in Banach Spaces

Abstract: A Banach space is rotund if the midpoint of each chord of the unit ball lies beneath the surface. In 1936 Clarkson [3] introduced the stronger notion of uniform rotundity. A Banach space is uniformly rotund if the midpoints of all chords of the unit ball whose lengths are bounded below by a positive number are uniformly buried beneath the surface. Since Clarkson's paper many authors have defined and studied geometric properties whose strengths lie between rotundity and uniform rotundity (see [1, 7-9,11, and 12]). Most of these properties can be classified as either localizations or directionalizations of uniform rotundity. The localizations--locally uniformly rotund and midpoint locally uniformly rotund-and the directionalizations--weakly uniformly rotund and uniformly rotund in every direction--have been of particular interest in the literature (see [4, 6, and 13]). In this paper six examples of Banach spaces are given that illustrate the distinctions among these generalizations of uniform rotundity as well as the independence of the localizations and directionalizations.

On the existence of weak solutions of differential equations in nonreflexive banach spaces

Abstract: THE STUDY of the Cauchy problem for differential equations in a Banach space relative to the strong topology has attracted much attention in recent years [l&4]. This study has taken two different directions. One direction is to impose compactness type conditions that guarantee only existence and the corresponding results are extensions of the classical Peano’s Theorem. The other approach is to utilize dissipative type conditions that assure existence and uniqueness of solutions, and the corresponding results are extensions of the classical Picard’s Theorem. However a similar study ofthe Cauchy problem in a Banach space relative to the weak topology has lagged behind. Recently Szep [4] proved Peano’s Theorem in the weak topology for differential equations in a reflexive Banach space. His main tools are the Eberlein-Smulian Theorem and the fact that a subset of a reflexive Banach space is weakly compact if and only if it is weakly closed and norm bounded. In this paper we wish to prove this theorem in arbitrary Banach spaces. Our first approach is to impose weak compactness type conditions in terms of the measure of weak noncompactness developed by De Blasi. We also impose weak dissipative type conditions and prove an existence and uniqueness theorem. Using these existence results and the partial orderings induced by cones, existence of extremal solutions and comparison results relative to the weak topology are also proved.

On the automorphisms of circular and Reinhardt domains in complex banach spaces

Abstract: The group Aut(D) of all biholomorphic automorphisms of a bounded circular domain D in a complex Banach space E is discussed. As an application the group Aut[D] is computed for the open unit balls of certain classical Banach spaces.

On the limit theorems for random variables with values in the spaces L p (2≦p<∞)

Abstract: We prove that whenever B is an infinite dimensional Banach space, there exists a B-valued random variable X failing the Central Limit Theorem (in short the CLT) and such that IE∥X∥2=∞ but yet satisfying the Law of the Iterated Logarithm (in short the LIL) We obtain a new sufficient condition for the LIL in Hilbert space and we characterize the random variables with values in l p or L p with 2

Existence theorems for Pareto optimization; multivalued and Banach space valued functionals

Abstract: ABSTRACr. Existence theorems are obtained for optimization problems where the cost functional takes values in an ordered Banach space. The order is defined in terms of a closed convex cone in the Banach space; and in this connection, several relevant properties of cones are studied and they are shown to coincide in the finite dimensional case. The notion of a weak (Pareto) extremum of a subset of an ordered Banach space is then introduced. Existence theorems are proved for extrema for Mayer type as well as Lagrange type problems-in a manner analogous to and including those with scalar valued cost. The side conditions are in the form of general operator equations on a class of measurable functions defined on a finite measure space. Needed closure and lower closure theorems are proved. Also, several analytic criteria for lower closure are provided. Before the appendix, several illustrative examples are given. In the appendix, a criterion (different from the one used in main text) is given and proved, for the Pareto optimality of an element.

On the Operator Identity ∑ AkXBk ≡ 0

Abstract: Let Aj and Bj (1 ≦ j ≦ m) be bounded operators on a Banach space ᚕ and let Φ be the mapping on , the algebra of bounded operators on ᚕ, defined by (1) We give necessary and sufficient conditions for Φ to be identically zero or to be a compact map or (in the Hilbert space case) for the induced mapping on the Calkin algebra to be identically zero. These results are then used to obtain some results about inner derivations and, more generally, about mappings of the form For example, it is shown that the commutant of the range of C(S, T) is “small” unless S and T are scalars.

On dualL 1-spaces and injective bidual Banach spaces

Abstract: In a previous paper (Israel J. Math.28 (1977), 313–324), it was shown that for a certain class of cardinals τ,l 1(τ) embeds in a Banach spaceX if and only ifL 1([0, 1]τ) embeds inX *. An extension (to a rather wider class of cardinals) of the basic lemma of that paper is here applied so as to yield an affirmative answer to a question posed by Rosenthal concerning dual ℒ1-spaces. It is shown that ifZ * is a dual Banach space, isomorphic to a complemented subspace of anL 1-space, and κ is the density character ofZ *, thenl 1(κ) embeds inZ *. A corollary of this result is that every injective bidual Banach space is isomorphic tol ∞(κ) for some κ. The second part of this article is devoted to an example, constructed using the continuum hypothesis, of a compact spaceS which carries a homogeneous measure of type ω1, but which is such thatl 1(ω1) does not embed in ℰ(S). This shows that the main theorem of the already mentioned paper is not valid in the case τ = ω1. The dual space ℰ(S)* is isometric to $$(L{}^1[0,1]^{\omega _1 } ) \oplus \left( {(\sum\limits_{\omega _1 } {{}^ \oplus L{}^1[0,1] \oplus l^1 (\omega _1 )} } \right)_1 ,$$ , and is a member of a new isomorphism class of dualL 1-spaces.

On semilinear evolution equations in Banach spaces.

Abstract: In this paper we shall present various conditions on A (t) and / sufficient for existence and/or uniqueness of mild or strict Solutions to problem (1); see Definition 3 and Definition 2 for the meaning of "mild" and "strict". We shall also look for implications between mild and strict Solutions. Concerning A(t), we shall assume that it generates an evolution System U (t, s), see Definition l, which may be "hyperbolic". The nonlinearity / will be continuous and additionally be compact or dissipative.

Simultaneous approximation in scales of Banach spaces

Abstract: The problem of verifying optimal approximation simultaneously in different norms in a Banach scale is reduced to verification of optimal approximation in the highest order norm. The basic tool used is the Banach space interpolation method developed by Lions and Peetre. Applications are given to several problems arising in the theory of finite element methods. 1. Introduction. In many papers concerning the mathematical analysis of finite element methods, certain approximation properties are assumed. In particular, it is often supposed that a given function may be approximated by a function in another space and that this approximation is "optimal" simultaneously in different norms. More precisely, let I2 be a bounded domain in R^ and Hs = ws2(£l) the Sobolev space of order s with norm ||-||s (cf. Lions and Magenes (9)). Let k and r be positive integers with k < r, and let {Sh' 0