Showing papers on "Banach space published in 1983"

A First Course in Functional Analysis

Abstract: Metric Space: 1.1 Definitions and examples 1.2 Inequalities of Holder and Minkowski 1.3 Examples continued $1_p$ spaces 1.4 Examples continued Function spaces 1.5 Convergence and related notions 1.6 Separable space, examples 1.7 Complete space, examples 1.8 Contractions, applications to differential and integral equations 1.9 Completion 1.10 Category, nowhere differentiable continuous functions 1.11 Compactness, continuity 1.12 Equicontinuity, application to differential equations 1.13 Stone-Weierstrass theorems 1.14 Normal families 1.15 Semi-continuity, application to arc length 1.16 Space of compact, convex sets Exercises Banach Spaces: 2.1 Vector space 2.2 Subspace 2.3 Quotient space 2.4 Dimension, Hamel basis 2.5 Algebraic dual, second dual 2.6 Convex sets 2.7 Ordered groups 2.8 Hahn-Banach theorem, separation form 2.9 Hahn-Banach theorem, extension form 2.10 Applications, Banach limits, invariant measure 2.11 Banach space, dual space 2.12 Hahn-Banach theorem in normed space 2.13 Uniform boundedness principle, applications 2.14 Lemma of F. Riesz, applications 2.15 Application to compact transformations 2.16 Applications, weak convergence, summability methods, approximate integration 2.17 Second dual space 2.18 Dual of $1_p$ 2.19 Dual of $C[a, b]$, Riesz representation theorem 2.20 Open mapping and closed graph theorems 2.21 Application, projections 2.22 Application, Schauder expansion 2.23 A theorem on operators in $C[0, 1]$ Exercises Measure and Integration, $L_p$ Spaces: 3.1 Lebesgue measure for bounded sets in $E_n$ 3.2 Lebesgue measure for unbounded sets 3.3 Totally $\sigma$ finite measures 3.4 Measurable functions, Egoroff theorem 3.5 Convergence in measure 3.6 Summable functions 3.7 Fatou and Lebesgue dominated convergence theorems 3.8 Integral as a set function 3.9 Signed measure, decomposition into measures 3.10 Absolute continuity and singularity of measures 3.11 The $L_p$ spaces, completeness 3.12 Approximation and smoothing operations 3.13 The dual of $L_p, p>1$ 3.14 The dual of $L_1$ 3.15 The individual ergodic theorem 3.16 $L_p$ convergence of Fourier series 3.17 Functions whose Fourier series diverge almost everywhere 3.18 Continuous functions which differ from all those having a given modulus Exercises Hilbert Space: 4.1 Inner product, Hilbert space 4.2 Basic lemma, projection theorem, dual 4.3 Application, mean ergodic theorem 4.4 Orthonormal sets, Fourier expansion 4.5 Application, isoperimetric theorem 4.6 Muntz theorem 4.7 Dimension, Riesz-Fischer theorem 4.8 Reproducing kernel 4.9 Application, Bergman kernel 4.10 Examples of complete orthonormal sets 4.11 Systems of Haar, Rademacher, Walsh applications Exercises Topological Vector Spaces: 5.1 Topology 5.2 Tychonoff theorem, application in Banach space 5.3 Topological vector space 5.4 Normable space 5.5 Space of measurable functions 5.6 Locally convex space 5.7 Metrizable space, space of entire functions 5.8 FK spaces 5.9 Application to summability methods 5.10 Ordered vector spaces 5.11 Banach lattice 5.12 Kothe spaces Exercises Banach Algebras: 6.1 Definition and examples 6.2 Adjunction of identity 6.3 Haar measure 6.4 Commutative Banach algebras, maximal ideals 6.5 The set $C(\scr{M})$ 6.6 Gelfand representation for algebras with identity 6.7 Analytic functions 6.8 Isomorphism theorem for algebras with identity 6.9 Algebras without identity 6.10 Application to $L_1(G)$ Exercises References Index.

Invariance Principles for Sums of Banach Space Valued Random Elements and Empirical Processes

Abstract: Almost sure and probability invariance principles are established for sums of independent not necessarily measurable random elements with values in a not necessarily separable Banach space. It is then shown that empirical processes readily fit into this general framework. Thus we bypass the problems of measurability and topology characteristic for the previous theory of weak convergence of empirical processes.

Contractivity in the numerical solution of initial value problems

Abstract: Consider a linear autonomous system of ordinary differential equations with the property that the norm |U(t)| of each solutionU(t) satisfies |U(t)|?|U(0)| (t?0). We call a numerical process for solving such a system contractive if a discrete version of this property holds for the numerical approximations. A givenk-step method is said to be unconditionally contractive if for each stepsizeh>0 the numerical process is contractive. In this paper a general theory is given which yields necessary and sufficient conditions for unconditional contractivity. It turns out that unconditionally contractive methods are subject to an order barrierp?1. Further the concept of a contractivity threshold is studied, which makes it possible to compare the contractivity behaviour of methods with an orderp>1 as well. Most theoretical results in this paper are formulated for differential equations in arbitrary Banach spaces. Applications are given to numerical methods for solving ordinary as well as partial differential equations.

Derivatives of analytic families of Banach spaces

Abstract: The theory of complex interpolation of Banach spaces and operators, which was developed by Calderon, Lions and S. G. Krein and extended by us and others, centers on the use of the maximum principle for analytic functions. In this paper we investigate the significance in interpolation theory of estimates for derivatives of analytic functions. We obtain norm estimates for certain linear and non-linear commutators and obtain new classical interpolation theorems. The paper has four sections. In the first section we show how Thorin's proof of the Riesz-Thorin theorem can be extended to give an estimate for a non-linear commutator and how analogous computations can be done in families of interpolation spaces. The second section introduces an extension of the complex interpolation theory which is designed to incorporate in a systematic way the type of estimates obtained in the first section. In the third section, the details of the abstract theory of the second section are filled in for several standard examples including LP spaces and weighted LP spaces. A brief final section contains some general comments and observations.

Strong Law of Large Numbers for Banach Space Valued Random Sets

Abstract: In this paper we prove a strong law of large numbers for random sets whose values are compact convex subsets of a Banach space.

Stability of isometries on Banach spaces

Abstract: Let X and Y be Banach spaces. A mapping /: X -» Y is called an E-isometry if | ||/(.x„) f(xi )|| \\x0 jc,|| | Y is called an e-isometry if | ||/(x0) — f(xx)\\ — \\xQ — xx\\ | Y there exists an isometry /: X -* y for which \\f(x) I(x)\\ < Ke for all x e X. This problem has been solved in a number of special cases (see [1] and [2] for a summary of such results), and Gruber [2, Theorem 1] went very far towards a general solution by showing that iff: X -» Y is a surjective e-isometry and /: X -» Y is an isometry for which 1(0) = /(0) and for which \\f(x) /(x)||/||x|| -» 0 uniformly as ||jc|| -» oo, then / is surjective and \\f(x) I(x)\\ < 5e for all x e X. In what follows we will show that such an isometry always exists so that the answer to the question of Hyers and Ulam is affirmative with K( X, Y) = 5 for all X and Y. We do this by establishing that: There exist constants A and B such that if /: X -» Y is a surjective e-isometry, then (1) ||/((*0 + xx)/2) (f(x0) +/(x,))/2|| < A(e\\x0 x,||)'/2 + Be

Limit Theorems for Sums of Weakly Dependent Banach Space Valued Random Variables

Abstract: We prove an estimate for the Prohorov-distance in the central limit theorem for strong mixing Banach space valued random variables. Using a recent variant of an approximation theorem of Berkes and Philipp (1979) we obtain as a corollary a strong invariance principle for absolutely regular sequences with error term $$t^{\tfrac{1}{2} - \gamma }$$ . For strong mixing sequences we prove a strong invariance principle with error term o((t log logt)1/2).

Topological Methods in Nonlinear Functional Analysis

Abstract: Contractors and fixed points by M. Altman The degree of mapping, and its generalizations by F. E. Browder Multiple fixed points of compact maps on wedgelike ANRS in Banach spaces by R. F. Brown The Nielsen numbers on surfaces by E. R. Fadell and S. Husseini A good class of eventually condensing maps by G. Fournier Iteration process for nonexpansive mappings by K. Goebel and W. A. Kirk The best approximation of bivariate functions of separable functions by M. von Golitschek and E. W. Cheney Positive solutions of operator equations in the nondifferentiable case by R. Guzzardi On fixed points of nonexpansive mappings by D. S. Jaggi Large oscillations of forced nonlinear differential equations by M. Martelli Fixed points and sequences of iterates in locally convex spaces by S. A. Naimpally, K. L. Singh, and J. H. W. Whitfield Fixed points theorems and Jung constant in Banach spaces by P. L. Papini Some results on multiple positive fixed points of multivalued condensing maps by W. V. Petryshyn Some problems and results in fixed point theory by S. Reich Contractive definitions revisited by B. E. Rhoades Fixed points, antipodal points and coincidences of n-acyclic multifunctions by H. Schirmer A coincidence theorem for topological vector spaces by V. M. Sehgal, S. P. Singh, and B. Watson Some random fixed point theorems by V. M. Sehgal and C. Waters.

Weak Pareto-optimal necessary conditions in a nondifferentiable multiobjective program on a banach space

Abstract: The nondifferentiable optimization theory with equality and inequality constraints is extended to a multiobjective program on a Banach space. We derive generalized conditions of the Fritz-John type given by Clarke's generalized gradient formula, which are necessary for weak Pareto-optimal solutions.

Spheres in infinite-dimensional normed spaces are Lipschitz contractible

Abstract: Let X be an infinite-dimensional normed space. We prove the following: (i) The unit sphere {x E X: II x II = I) is Lipschitz contractible. (ii) There is a Lipschitz retraction from the unit ball of X onto the unit sphere. (iii) There is a Lipschitz map T of the unit ball into itself without an approximate fixed point, i.e. inf II x TxI 1: 11 x 11 ) I > 0. Introduction. Let X be a normed space, and let Bx= {x E X: Ix I 1} and Sx = {x E X: 1I x II = 1 } be its unit ball and unit sphere, respectively. Brouwer's fixed point theorem states that when X is finite dimensional, every continuous self-map of Bx admits a fixed point. Two equivalent formulations of this theorem are the following. 1. There is no continuous retraction from Bx onto Sx. 2. Sx is not contractible, i.e., the identity map on Sx is not homotopic to a constant map. It is well known that none of these three theorems hold in infinite-dimensional spaces (see e.g. [1]). The natural generalization to infinite-dimensional spaces, however, would seem to require the maps to be uniformly-continuous and not merely continuous. Indeed in the finite-dimensional case this condition is automatically satisfied. In this article we show that the above three theorems fail, in the infinite-dimensional case, even under the strongest uniform-continuity condition, namely, for maps satisfying a Lipschitz condition. More precisely, we prove THEOREM. Let X be an infinite-dimensional normed space. Then (1) The unit sphere Sx is Lipschitz contractible. (2) There is a Lipschitz retraction from Bx onto Sx. (3) There is a Lipschitz map T: Bx -Bx without an approximate fixed point, i.e. inf{IIx TxII: x E Bx} = d > 0. The first study of Lipschitz maps without approximate fixed points, and Lipschitz retractions from Bx onto Sx, was done by K. Goebel [3]. B. Nowak [5] proved the theorem for several classical Banach spaces. Our work was greatly influenced by the work of Nowak. Actually, the general scheme of the proof as well as two of the three Received by the editors November 1, 1982. 1980 Mathematics Subject Classification. Primary 47H 10, 46B20. 1 The first author was partially supported by the Fund for the Promotion of Research at the Technion -Israel Institute of Technology. 01983 American Mathematical Society 0002-9939/82/0000-1337/$02.50

On the structure of the solution set for differential inclusions in a banach space

Abstract: In this article a differential inclusion is considered, where the mapping takes values in the family of all nonempty compact convex subsets of a Banach space, is upper semicontinuous with respect to for almost every , and has a strongly measurable selection for every . Under certain compactness conditions on proofs are given for a theorem on the existence of solutions, a theorem on the upper semicontinuous dependence of solutions on the initial conditions, and an analogue of the Kneser-Hukuhara theorem on connectedness of the solution set.Bibliography: 20 titles.

Norm attaining operators and renormings of Banach spaces

Abstract: We define two geometric concepts of a Banach space, property α and β, which generalize in a certain way the geometric situation ofl andc o. These properties have been used by J. Lindenstrauss and J. Partington in the study of norm attaining operators. J. Partington has shown that every Banach space may (3+e)-equivalently be renormed to have property β. We show that many Banach spaces (e.g., every WCG space) may (3+e)-equivalently be renormed to have property α. However, an example due to S. Shelah shows that not every Banach space is isomorphic to a Banach space with property α.

On homogeneous polynomials on a complex ball

Abstract: We prove that there exist «-homogeneous polynomials p„ on a complex d-dimensional ball such that II /»„ II „o = 1 and Ily7„ll2 * fn2~d. This enables us to answer some questions about Hp and Bloch spaces on a complex ball. We also investigate interpolation by «-homogeneous polynomials on a 2-dimensional complex ball. Introduction. The starting point of our investigation was a question asked by S. Waigner: Is the identity map from Hx(Bd) into Hx(Bd) (Bd is a unit ball in Cd), d > 1, a compact linear map. This question has a connection with the well-known open problem (cf. [5]): does there exist a nonconstant inner function on Bd, d > l?1 The existence of such an inner function would imply our Corollary 1.5, namely that this operator is not compact. We obtain this result by exhibiting «-homogeneous polynomials pn which in some respects resemble the monomials z" in the one-dimensional case (Theorem 1.2). We give two proofs of this theorem, one in §1 and the other at the end of §2. Those polynomials enable us to also answer a question of R. Timoney (Corollary 1.9). We hope that they will find some other applications. In §2 we investigate interpolating «-homogeneous polynomials on the unit sphere in two-dimensional complex space. The motivation for this study is the following well-known open problem (cf. [6]): does there exist a function 1, such that for every finite-dimensional Banach space X we have d(X, l^nX) = 1}. The point (1,0,... ,0) G Sd will be denoted by 1. On Sd we have the natural rotation-invariant probability measure a. Wn(Sd) will denote the space of all «-homogeneous polynomials on Cd restricted to Sd. On Wn(Sd) we will consider various norms. For f E Wn(Sd) we put 11/11^ = ilsd\fWd°iS)r/p if 1 " we obtain A/(l) = (P„f)(l)=f(i), so X = 1. This means that Pn is an orthogonal projection onto W2(Sd). Its norm as an operator on LJSd) equals aid, n)a(d,2n)-x xfn2~d. Before we start the proof of Theorem 1.2 let us recall some well-known notions concerning finite-dimensional Banach spaces. Let X, Y be finite-dimensional Banach spaces. We put X(*) = infill Til ■ \\S\\:X^Lxtx, TS = Id*, Lx is an arbitrary Ljuy-spacej and d(X, Y) = inf{||71 • lir-'ll: T: X -> Yis 1-1 and onto}. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use HOMOGENEOUS POLYNOMIALS ON A COMPLEX BALL 109 The number A(A') is called a projection constant of X and d(X, Y) is called the Banach-Mazur distance between X and Y. It is well known and easy to see that (1) X(X) ^/«~. More information on this can be found in [6] and [4,22.1 and 28.1]. We will also use the following two lemmas. Lemma 1.3. inf{\\p\\2: p E W„(Sd) and H/HI«, = 1) = ya(d,2n). Proof. Since the spaces under consideration are finite dimensional, there exists a polynomial p0 realising the minimum. We can assume p0ii) — 1 = ll/Poll«,It is easily seen that Pin = -^r---f%o(h,ei9iz2,...,ea^zd)der-'ded_l = (S,ir. (2ir) •'o Jo Moreover, ||p\\2 « ||p0\\2 and \\p\\x = IIpQIIx = 1. Sincep0 was minimal we infer Il PII2 = II Po II2This proves the lemma. Lemma 1.4. dimW„(Sd) = a(d,2n)~x. Proof. It follows from Proposition 1.1 that dim Wn(Sd) = trace Pn. Since Pn is a projection its trace equals the square of its Hubert-Schmidt norm, which, as is well known, equals a(d,2n)-2f j \(S,V)\2"da(!;)da(Ti) = a(d,2n)-1. Now we are ready to prove the theorem. Proof of Theorem 1.2. We apply (1) for X = W2(Sd) and Y = Wn°°(Sd). Using (2), Proposition 1.1 and Lemma 1.4 we obtain (3) d{W2(Sd), W?iSd)) ^ X{W2(Sd))XiW-(Sd))-X > 2x~d(fr/2)]ia(d,2ny] . Let / denote the identity map from WTM(Sd) into W2(Sd). By the definition of the Banach-Mazur distance we have 11/II ■ III~x\\ > d(W2(Sd),WTM(Sd)). Lemma 1.3 gives || 7"11| = ]Ja(d, 2«)~ so by (3) we have \\I\\^2x'd^/2. This inequality proves the theorem. Remark. Actually d(W2(Sd), W„°°(Sd)) = \\I\\ ■ \\rx\\. This follows from [1, Lemma 4.6]. Corollary 1.5. id: Hx(Bd) -» Hx(Bd) is not a compact operator. Proof. Polynomials pn exhibited in Theorem 1.2 are orthogonal. Moreover by the Schwartz inequality, fW2~d tr4-d. This implies that id: Hx(Bd) -» Hx(Bd) is not a compact operator. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 110 J. RYLL AND P. WOJTASZCZYK Proposition 1.6. Let (nk) be a lacunary sequence of natural numbers and let p„k E W„k(Sd) be such that \\p„k\\x = 1 and II/>„J|, > c, k = 0,1,2.... Then there exist constants A and B such that (4) A-X(i\ak\2)l/1 <\\lakP„kj

A construction of ℒ∞-spaces and related Banach spacesand related Banach spaces

Abstract: Let λ>1. We prove that every separable Banach space E can be embedded isometrically into a separable ℒ∞λ-spaceX such thatX/E has the RNP and the Schur property. This generalizes a result in [2]. Various choices ofE allow us to answer several questions raised in the literature. In particular, takingE = l2, we obtain a ℒ∞λ-spaceX with the RNP such that the projective tensor product\(X\hat \otimes X\) containsc0 and hence fails the RNP. TakingE=L1, we obtain a ℒ∞λ-space failing the RNP but nevertheless not containingc0.

Degree of mapping for nonlinear mappings of monotone type

Abstract: A classical degree function is constructed for pseudomonotone mappings from a reflexive Banach space to its dual, using Galerkin approximations. This generalizes the Leray-Schauder degree when the Banach space is a Hilbert space and yields a flexible analytical tool for the study of nonlinear elliptic problems of higher order in divergence form.

Central limit theorems and weak laws of large numbers in certain banach spaces

Abstract: For B a type 2 Banach lattice, we obtain a relationship between the central limit theorem in B and the weak law of large numbers (for the sum of the squares of the random vectors) in another Banach lattice B(2). We then obtain some two-sided estimates for E∥Sn∥pwhich in lpspaces, 1≦p<∞, give n.a.s.c. for the weak law of large numbers. As a consequence of these estimates we also solve the domain of attraction problem in lp, p<2. Several examples and counterexamples are provided.