Showing papers on "Lp space published in 1978"
Book•
01 Jan 1978
TL;DR: In this paper, the spectral theory of linear operators in normed spaces and their spectrum has been studied in the context of bounded self-and-adjoint linear operators and their applications in quantum mechanics.
Abstract: Metric Spaces. Normed Spaces Banach Spaces. Inner Product Spaces Hilbert Spaces. Fundamental Theorems for Normed and Banach Spaces. Further Applications: Banach Fixed Point Theorem. Spectral Theory of Linear Operators in Normed Spaces. Compact Linear Operators on Normed Spaces and Their Spectrum. Spectral Theory of Bounded Self--Adjoint Linear Operators. Unbounded Linear Operators in Hilbert Space. Unbounded Linear Operators in Quantum Mechanics. Appendices. References. Index.
2,781 citations
01 Feb 1978
TL;DR: In this article, the authors give an answer to Ulam's problem: "Give conditions in order for a linear mapping near an approximately linear mapping to exist", and prove it for the case n = 1.
Abstract: Let E1, E2 be two Banach spaces, and let f: E1 -* E2 be a mapping, that is "approximately linear". S. M. Ulam posed the problem: "Give conditions in order for a linear mapping near an approximately linear mapping to exist". The purpose of this paper is to give an answer to Ulam's problem. THEOREM. Consider E1, E2 to be two Banach spaces, and let f: E1 -> E2 be a mapping such that f (tx) is continuous in t for each fixed x. Assume that there exists 0 > 0 andp E [0, 1) such that IIf(x + y) f (x) f(A)lI 0. The verification of (3) follows by induction on n. Indeed the case n = 1 is clear because by the hypothesis we can find 0, that is greater or equal to zero, andp such that 0 < p < 1 with 11[f(2x)]/2 -f(x)ll (4) IIxIIp Assume now that (3) holds and we want to prove it for the case (n + 1). However this is true because by (3) we obtain II [f (2n 2x)]/2 n f(2x)llI nI I*2x)2P < m E 2m(P therefore Received by the editors December 1, 1977. AMS (MOS) subject classifications (1970). Primary 47H15; Secondary 39A15.
2,694 citations
Journal Article•
TL;DR: In this paper, the conditions générales d'utilisation (http://www.compositio.org/conditions) of the agreement with the Foundation Compositio Mathematica are described.
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.
182 citations
143 citations
126 citations
107 citations
Journal Article•
TL;DR: In this article, Sijthoff and Noordhoff showed that X need not be locally convex, and that X is not a B-convex Banach space.
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
101 citations
100 citations
01 Mar 1978
TL;DR: In this article, the classical moment inequality that the Lp norm of Ai is at most p/(p - 1) times the 4L norm of,u is shown to be sharp.
Abstract: Partial order the set of distributions on the real line by v 1, the classical moment inequality that the Lp norm of Ai (and of ,u*) is at most p/(p - 1) times the 4L norm of ,u is shown to be sharp.
TL;DR: 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.
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.
Journal Article•
TL;DR: In this paper, the authors present various conditions on A (t) and / sufficient for existence and uniqueness of mild or strict solutions to problem (1); see Definition 3 and Definition 2 for the meaning of "mild" and "strict".
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.
TL;DR: In this paper, an approach to the local theory of Banach spaces via the ultrapower construction is presented, which includes a detailed study of ultrapowers and their dual spaces as well as a definition of a new notion, the notion of u-extension of a Banach space.
Abstract: ABssAcr. The present paper is an approach to the local theory of Banach spaces via the ultrapower construction. It includes a detailed study of ultrapowers and their dual spaces as well as a definition of a new notion, the notion of a u-extension of a Banach space. All these tools are used to give a unified definition of many classes of Banach spaces characterized by local properties (such as the EP-spaces). Many examples are given; also, as an application, it is proved that any e -space, 1 < p < oo, has an ultrapower which is isomorphic to an Lp-space.
TL;DR: In this article, 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 in the Banach space interpolation method developed by Lions and Peetre.
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
TL;DR: In this article, a fixed point theorem for 2-metric spaces is established and some interesting particular cases of this theorem are also obtained. But this theorem is not applicable to 2-dimensional spaces.
Abstract: In this paper we establish a fixed point theorem for 2-metric spaces. Some interesting particular cases of this theorem are also obtained.
TL;DR: In this article, a complete analysis of the relationship between the existence of bounded point evaluation and the density of N(E) in Lp(E), 1 ⩽ p < ∞ is presented.
TL;DR: In this paper, it was shown that with an infinitely divisible random variable with values in a separable Banach space, there is a Levy-Khintchine formula.
TL;DR: In this article, the authors generalize SteCkin's result to a wider class of Banach spaces and show that this class of spaces is self-dual, it contains all uniformly convex spaces, and there exists a subspace G which is B-isomorphic to F and is almost Chebyshev.
TL;DR: In this article, relations between BESOV spaces B8∞,q, HOLDER spaces C8, ZYGMUND spaces ˆ8, dual spaces of the spaces F81,q and the space of bounded mean oscillation BMO, the space L∞.
Abstract: The paper deals with relations between the following types of spaces: of BESOV spaces B8∞,q, HOLDER spaces C8, ZYGMUND spaces ˆ8, dual spaces of the spaces F81,q, the space of bounded mean oscillation BMO, the space L∞. The following properties are considered: Inclusions, interpolation, duality, multipliers, and equivalent norms.