APPROXIMATE ISOMETRIES ON FINITE-DIMENSIONAL NORMED SPACES

Size: px
Start display at page:

Download "APPROXIMATE ISOMETRIES ON FINITE-DIMENSIONAL NORMED SPACES"

Transcription

1 APPROXIMATE ISOMETRIES ON FINITE-DIMENSIONAL NORMED SPACES S. J. DILWORTH Abstract. Every ε-isometry u between real normed spaces of the same finite dimension which maps the origin to the origin may by uniformly approximated to within 2ε by a linear isometry. Under a smoothness hypothesis, necessary and sufficient conditions are obtained for the same conclusion to hold for a given ε-isometry between infinite-dimensional Banach spaces. 1. Introduction The following notion of an approximate isometry between Banach spaces was introduced by Hyers and Ulam [5]. Definition 1. Let 0 <ε< and let E and F be real Banach spaces. A mapping f : E F is said to be an ε-isometry if f(x) f(y) x y ε (x, y E). Hyers and Ulam proved that every surjective ε-isometry on a Hilbert space may be uniformly approximated by an affine isometry. Recently [1] Bhatia and Semrl proved that every (not necessarily surjective) ε-isometry f from arealn-dimensional Euclidean space E n into itself, for which f(0) = 0, may be uniformly approximated to within 2ε by a linear isometry. The main purpose of this paper is to show that the same result holds for arbitrary finite-dimensional normed spaces, thus answering the question raised at the end of [1]. 2. Results Let E be a normed space. Recall that the norm of E is said to be Fréchet differentiable at y E if there exists a functional y E such that (1) y + h =1+y (h)+ h δ(h) (h E), where δ(h) 0as h 0. In fact y will be the unique norming functional for y. We shall write x y if y (x) = 0 (N.B. x y does not imply y x). Suppose that (x k ) k=1 is a bounded sequence in E (i.e. x k Cfor some fixed C) such that x k u for each k and that (λ k ) k=1 is a sequence of 1991 Mathematics Subject Classification. 46B04. 1

2 2 S. J. DILWORTH nonzero real numbers. The homogeneity of the norm function and (1) imply that λ k y + x k = λ k y +(1/λ k )x k = λ k (1 + ( x k / λ k )δ((1/ λ k )x k ) = λ k + x k δ((1/ λ k )x k ). In particular, if λ k as k, then (since x k C)wehave (2) λ k y + x k λ k 0ask. In the following proposition the hypothesis asserts that the set of points in E at which the norm is Fréchet differentiable should be dense. Among others, this hypothesis is satisfied by the Asplund spaces, which are by definition the spaces E for which every continuous convex function defined on an open subset U of E is Fréchet differentiable on a dense G δ subset of U (see e.g. [7, p.14]). It is known that a Banach space E is an Asplund space if and only if E has the Radon-Nikodým Property (equivalently, if and only if every separable subspace of E has a separable dual space) [7, p.34]. In particular, all reflexive Banach spaces are Asplund spaces. The hypothesis of the proposition is also satisfied by many spaces which are not Asplund spaces,e.g.byl. However, it is not satisfied by the important spaces l 1, L 1 [0, 1] and L [0, 1]. The existence of a dense set of points of Fréchet differentiability of the norm permits the argument of [1, p.503] to be generalized as follows. Proposition 1. Let E and F be normed spaces and suppose that the the set of points in E at which the norm is Fréchet differentiable is dense. Let f : E F be an ε-isometry, with f(0) = 0, for which there exist a constant K and a surjective linear isometry u : E F such that f(x) u(x) K for all x E. Then f(x) u(x) 2ε (x E). Proof. By replacing f by u 1 f, we may suppose without any loss of generality that F = E and that u = Id (the identity function), so that f(x) x Kfor all x E. Let x E and suppose that f(x) x = a 0. Fixα>0. By assumption there exists a unit vector y at which the norm is Fréchet differentiable such that f(x) x = ay + z, where z <α. Write x = x 0 + by (b R) withx 0 y. For every positive integer m we have f(x + my) =x+my + v m, where v m Kby hypothesis. Write v m = b m y + u m (b m R) withu m y. Let y be the unique norming functional at the point y. Then b m y = b m = y (b m y+u m ) = y (v m ) v m K, and so by the triangle inequality u m b m y + v m 2K. The fact that f is an ε-isometry with f(0) = 0 gives f(x + my) x+my ε,

3 APPROXIMATE ISOMETRIES ON FINITE-DIMENSIONAL NORMED SPACES 3 which can be rewritten as (m + b + b m )y +(x 0 +u m ) (m+b)y+x 0 ε. Since u m 2K, and since both x 0 y and u m y, the remarks before this lemma (in particular (2)) yield lim sup m+b+b m m+b ε. Thus lim sup b m ε. Using again the fact that f is an ε-isometry we have m ε f(x+my) f(x) m+ε, i.e. m ε (my + v m )+(x f(x)) m+ε, which can be rewritten as m ε (m a+b m )y+u m z m+ε. Since z <αthe triangle inequality gives m ε α< (m a+b m )y+u m <m+ε+α. Since u m 2Kand u m y, it follows from (2) that this norm can be made as close to m a + b m as we please by taking m sufficiently large. Since lim sup b m εit follows that a 2ε + α. Finally, since α can be chosen arbitrarily small, it follows that a 2ε. Remark 1. The surjectivity of u is essential. To see this, fix M > 0and ε>0 and consider f : R R 2 (with the Euclidean norm) given by { (x, 0) for x 0, f(x) = (x, M(1 e cx )) for x 0. Then f is an ε-isometry provided c = c(m,ε) is sufficiently small. However, sup{ f(x) (x, 0) : x R} = M, and so for large M the conclusion of the Proposition is not valid for the (non-surjective) isometry u(x) =(x, 0). Theorem 1. Let E and F be real normed spaces of the same finite dimension. Let f : E F be an ε-isometry from E into F with f(0) = 0. There exists a unique linear isometry u : E F such that u(x) f(x) 2ε (x E). Proof. The uniqueness of the linear isometry u is obvious and so it suffices to prove the existence of u. First, by a result attributed to P. L. Renz [2, Lemma 2.8], there exists a continuous 4ε-isometry g : E F such that g(0) = 0 and g(x) f(x) 2ε(x E). Secondly, by a result attributed to Dallas Webster [2, Prop. 4.1], whose proof contains an elegant application of the Borsuk-Ulam Theorem on antipodal points, every continuous approximate isometry between two normed spaces of the same finite dimension is surjective. Thus, g is a continuous surjective 4ε-isometry with g(0) = 0. By a result of Gruber [4, Theorem 3] there exists a surjective linear isometry u : E F such that u(x) g(x) 12ε. It follows from the triangle inequality that f(x) u(x) f(x) g(x) + g(x) u(x) 2ε+12ε=14ε.

4 4 S. J. DILWORTH As observed above, every finite-dimensional normed space satisfies the hypothesis of Proposition 1. In fact, it is a classical result that every continuous convex function on R n is differentiable almost everywhere with respect to n-dimensional Lebesgue measure (see e.g. [8, 7, Th. 25.5]). Finally, an application of Proposition 1 with K = 14ε yields the desired conclusion. Remark 2. It was shown in [1] that the upper bound of 2ε is best possible even in the case E = F = R. Here is a simple example; define f ε : R R thus: { x ε for x/ {0,ε}, f ε (x) = x for x {0,ε}. Clearly, f ε is a (surjective) ε-isometry, with f ε (0) = 0, and the identity function is the unique linear isometry which approximates f ε uniformly. But max{ f ε (x) x : x R} = f ε (ε) ε =2ε. The reader is referred to [6] for an example of a homeomorphism of R 2 for which the bound of 2ε is attained. An example is given in [5] (also [1]) which shows that the assumption that E and F should have the same dimension cannot be removed, even for Euclidean spaces, and that the result cannot be extended to infinite-dimensional spaces without some additional assumption. (See Theorem 2 below for a result of this kind.) Remark 3. For surjective ε-isometries Theorem 1 was proved by Gruber [4, Theorem 3] with the estimate 5ε. Hyers and Ulam [5] asked (implicitly) whether every surjective ε-isometry between normed spaces may be uniformly approximated by an isometry to within an error of Kε. This long-standing open question was finally answered positively by Gevirtz [3] with K = 5. Recently, Omladi c and Semrl [6] obtained the estimate K =2, which for linear approximation is optimal (see Remark 2 above). Corollary 1. Let E and F be real normed spaces of the same finite dimension and let f : E F be an ε-isometry from E into F. There exists an affine isometry v : E F such that v(x) f(x) 2ε (x E). Proof. Let g(x) = f(x) f(0) (x E). Then g is an ε-isometry with g(0) = 0. Theorem 1 applied to g gives a linear isometry u for which the affine isometry v(x) = u(x)+ f(0) is the required approximation to f. Some more terminology must now be introduced. A closed ball in a normed space with centre x and radius r will be denoted B(x, r). Recall that the Hausdorff distance between two subsets A and B of a metric space is defined by d H (A, B) =inf{r 0:A (B) r,b (A) r }, where (A) r = {x : d(x, A) <r}.letδ>0. A set S E will be said to be δ-dense in E if for every x E there exists s S such that x s δ;a mapping f : E F will be said to be δ-onto if f(e) isδ-dense in F.

5 APPROXIMATE ISOMETRIES ON FINITE-DIMENSIONAL NORMED SPACES 5 The next result is an infinite-dimensional version of Theorem 1. Under the hypothesis of Proposition 1 it provides necessary and sufficient conditions for the possibility of approximating an ε-isometry uniformly to within 2ε. Theorem 2. Let E and F be Banach spaces and suppose that the set of points in E at which the norm is Fréchet differentiable is dense. Let f : E F be an ε-isometry with f(0) = 0. Then the following are equivalent: (1) there exists a surjective linear isometry u : E F such that u(x) f(x) 2ε (x E); (2) f is δ-onto for some δ>0; (3) d H (f(e),f) <. Proof. Clearly, (2) and (3) are equivalent, and so it suffices to prove the equivalence of (1) and (2). Obviously, (1) (2) with δ =2ε. Suppose that (2) holds. Then the linear span of f(e) isdenseinf, which implies that the cardinality of E is not less than the cardinality of F. Let S = {x α : α A} Xbe a maximal set containing 0 satisfying the condition x α x β 2(α β). Then (2) and the fact that f is an ε-isometry readily imply that f(s) is(2+ε+δ)-dense in F. For each α Athere exists a surjection g α : B(x α, 1/2) B(f(x α ), 2+ε+δ) withg α (x α )=f(x α ). Define g : E F thus: { g α (x) for x B(x α,1/2), g(x) = f(x) otherwise. Then g is surjective and g(0) = 0. For x B(x α, 1/2), we have g(x) f(x) g α (x α ) f(x α ) + f(x α ) f(x) (2 + ε + δ)+(1/2+ε). Thus g(x) f(x) 5/2+2ε+δ (x E), and hence g is a (5 + 5ε +2δ)- isometry from E onto F. By a theorem of Gevirtz [3], there exists a surjective linear isometry u : E F with u(x) g(x) 5(5 + 5ε +2δ)(x E), which implies that u(x) f(x) K(x E) fork = 55/2 + 27ε + 11δ. Finally, (1) now follows from Proposition 1. The author does not know whether Theorem 2 holds without the hypothesis on E. For general Banach spaces, however, the following result affords necessary and sufficient conditions for the existence of a linear isometry which uniformly approximates a given ε-isometry. Proposition 2. Let f : E F be an ε-isometry between Banach spaces with f(0) = 0. The following are equivalent: (1) There exist K>0and a linear isometry u from E into F such that u(x) f(x) K (x E); (2) There exist M>0andaclosedsubspaceF 1 of F such that d H (f(e),f 1 )<M<.

6 6 S. J. DILWORTH Proof. If (1) holds then d H (u(e),f(e)) K, and so (2) holds with F 1 = u(e) andm=k. Suppose that (2) holds. Refining the proof of Theorem 2 (the details are omitted) one can construct a surjective mapping g : E F 1 satisfying g(0) = 0 and g(x) f(x) M+2ε (x E). Then g is a (2M +5ε)-isometry, and so by [6, Theorem 1] there exists a linear isometry from E onto F 1 satisfying u(x) g(x) 4M+10ε (x E). It follows that f satisfies (1) with K(M,ε) =5M+12ε. Remark 4. The example after Proposition 1 shows that (when F 1 F )the dependence of K(M,ε) onmcannot be eliminated and is at least linear in M. Acknowledgement. The author would like to thank Ralph Howard, Jim Roberts, Anton Schep and the members of the University of South Carolina Functional Analysis Seminar for comments which led to a simplification of the proof of Theorem 1. References [1] Rajendra Bhatia and Peter Semrl, Approximate Isometries on Euclidean Spaces, Amer. Math. Monthly, 104, 1997, [2] R. D. BOURGIN, Approximate isometries on finite-dimensional Banach spaces, Trans. Amer. Math. Soc. 207 (1975) [3] Julian Gevirtz, Stability of Isometries on Banach Spaces, Proc. Amer. Math. Soc., 89, 1983, [4] P.M. Gruber, Stability of Isometries, Trans. Amer. Math. Soc., 245, 1978, [5] D.H. Hyers and S.M. Ulam, On approximate isometries, Bull. Amer. Math. Soc., 51, 1945, [6] M. Omladi c andp. Semrl On non linear perturbations of isometries, Math. Ann., 303, 1995, [7] Robert R. Phelps, Convex functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, Vol. 1364, Springer-Verlag, Berlin Heidelberg, [8] R. Tyrell Rockafellar, Convex Analysis, Princeton University Press, Princeton, New Jersey, Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A. address: dilworth@math.sc.edu

SOME PROPERTIES ON THE CLOSED SUBSETS IN BANACH SPACES

SOME PROPERTIES ON THE CLOSED SUBSETS IN BANACH SPACES ARCHIVUM MATHEMATICUM (BRNO) Tomus 42 (2006), 167 174 SOME PROPERTIES ON THE CLOSED SUBSETS IN BANACH SPACES ABDELHAKIM MAADEN AND ABDELKADER STOUTI Abstract. It is shown that under natural assumptions,

More information

Functional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...

Functional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability... Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................

More information

Hyers-Ulam constants of Hilbert spaces

Hyers-Ulam constants of Hilbert spaces Hyers-Ulam constants of Hilbert spaces Taneli Huuskonen and Jussi Väisälä Abstract The best constant in the Hyers-Ulam theorem on isometric approximation in Hilbert spaces is equal to the Jung constant

More information

The small ball property in Banach spaces (quantitative results)

The small ball property in Banach spaces (quantitative results) The small ball property in Banach spaces (quantitative results) Ehrhard Behrends Abstract A metric space (M, d) is said to have the small ball property (sbp) if for every ε 0 > 0 there exists a sequence

More information

NUMERICAL RADIUS OF A HOLOMORPHIC MAPPING

NUMERICAL RADIUS OF A HOLOMORPHIC MAPPING Geometric Complex Analysis edited by Junjiro Noguchi et al. World Scientific, Singapore, 1995 pp.1 7 NUMERICAL RADIUS OF A HOLOMORPHIC MAPPING YUN SUNG CHOI Department of Mathematics Pohang University

More information

Boundedly complete weak-cauchy basic sequences in Banach spaces with the PCP

Boundedly complete weak-cauchy basic sequences in Banach spaces with the PCP Journal of Functional Analysis 253 (2007) 772 781 www.elsevier.com/locate/jfa Note Boundedly complete weak-cauchy basic sequences in Banach spaces with the PCP Haskell Rosenthal Department of Mathematics,

More information

An introduction to some aspects of functional analysis

An introduction to some aspects of functional analysis An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms

More information

Subdifferential representation of convex functions: refinements and applications

Subdifferential representation of convex functions: refinements and applications Subdifferential representation of convex functions: refinements and applications Joël Benoist & Aris Daniilidis Abstract Every lower semicontinuous convex function can be represented through its subdifferential

More information

Ginés López 1, Miguel Martín 1 2, and Javier Merí 1

Ginés López 1, Miguel Martín 1 2, and Javier Merí 1 NUMERICAL INDEX OF BANACH SPACES OF WEAKLY OR WEAKLY-STAR CONTINUOUS FUNCTIONS Ginés López 1, Miguel Martín 1 2, and Javier Merí 1 Departamento de Análisis Matemático Facultad de Ciencias Universidad de

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journal of Inequalities in Pure and Applied Mathematics ON SOME APPROXIMATE FUNCTIONAL RELATIONS STEMMING FROM ORTHOGONALITY PRESERVING PROPERTY JACEK CHMIELIŃSKI Instytut Matematyki, Akademia Pedagogiczna

More information

THE GEOMETRY OF CONVEX TRANSITIVE BANACH SPACES

THE GEOMETRY OF CONVEX TRANSITIVE BANACH SPACES THE GEOMETRY OF CONVEX TRANSITIVE BANACH SPACES JULIO BECERRA GUERRERO AND ANGEL RODRIGUEZ PALACIOS 1. Introduction Throughout this paper, X will denote a Banach space, S S(X) and B B(X) will be the unit

More information

THE NEARLY ADDITIVE MAPS

THE NEARLY ADDITIVE MAPS Bull. Korean Math. Soc. 46 (009), No., pp. 199 07 DOI 10.4134/BKMS.009.46..199 THE NEARLY ADDITIVE MAPS Esmaeeil Ansari-Piri and Nasrin Eghbali Abstract. This note is a verification on the relations between

More information

A Note on the Class of Superreflexive Almost Transitive Banach Spaces

A Note on the Class of Superreflexive Almost Transitive Banach Spaces E extracta mathematicae Vol. 23, Núm. 1, 1 6 (2008) A Note on the Class of Superreflexive Almost Transitive Banach Spaces Jarno Talponen University of Helsinki, Department of Mathematics and Statistics,

More information

ADJOINTS, ABSOLUTE VALUES AND POLAR DECOMPOSITIONS

ADJOINTS, ABSOLUTE VALUES AND POLAR DECOMPOSITIONS J. OPERATOR THEORY 44(2000), 243 254 c Copyright by Theta, 2000 ADJOINTS, ABSOLUTE VALUES AND POLAR DECOMPOSITIONS DOUGLAS BRIDGES, FRED RICHMAN and PETER SCHUSTER Communicated by William B. Arveson Abstract.

More information

Exercise Solutions to Functional Analysis

Exercise Solutions to Functional Analysis Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n

More information

Analysis III Theorems, Propositions & Lemmas... Oh My!

Analysis III Theorems, Propositions & Lemmas... Oh My! Analysis III Theorems, Propositions & Lemmas... Oh My! Rob Gibson October 25, 2010 Proposition 1. If x = (x 1, x 2,...), y = (y 1, y 2,...), then is a distance. ( d(x, y) = x k y k p Proposition 2. In

More information

arxiv: v1 [math.fa] 2 Jan 2017

arxiv: v1 [math.fa] 2 Jan 2017 Methods of Functional Analysis and Topology Vol. 22 (2016), no. 4, pp. 387 392 L-DUNFORD-PETTIS PROPERTY IN BANACH SPACES A. RETBI AND B. EL WAHBI arxiv:1701.00552v1 [math.fa] 2 Jan 2017 Abstract. In this

More information

CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES. Jong Soo Jung. 1. Introduction

CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES. Jong Soo Jung. 1. Introduction Korean J. Math. 16 (2008), No. 2, pp. 215 231 CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES Jong Soo Jung Abstract. Let E be a uniformly convex Banach space

More information

Overview of normed linear spaces

Overview of normed linear spaces 20 Chapter 2 Overview of normed linear spaces Starting from this chapter, we begin examining linear spaces with at least one extra structure (topology or geometry). We assume linearity; this is a natural

More information

Epiconvergence and ε-subgradients of Convex Functions

Epiconvergence and ε-subgradients of Convex Functions Journal of Convex Analysis Volume 1 (1994), No.1, 87 100 Epiconvergence and ε-subgradients of Convex Functions Andrei Verona Department of Mathematics, California State University Los Angeles, CA 90032,

More information

2. Function spaces and approximation

2. Function spaces and approximation 2.1 2. Function spaces and approximation 2.1. The space of test functions. Notation and prerequisites are collected in Appendix A. Let Ω be an open subset of R n. The space C0 (Ω), consisting of the C

More information

(convex combination!). Use convexity of f and multiply by the common denominator to get. Interchanging the role of x and y, we obtain that f is ( 2M ε

(convex combination!). Use convexity of f and multiply by the common denominator to get. Interchanging the role of x and y, we obtain that f is ( 2M ε 1. Continuity of convex functions in normed spaces In this chapter, we consider continuity properties of real-valued convex functions defined on open convex sets in normed spaces. Recall that every infinitedimensional

More information

arxiv:math/ v1 [math.fa] 21 Mar 2000

arxiv:math/ v1 [math.fa] 21 Mar 2000 SURJECTIVE FACTORIZATION OF HOLOMORPHIC MAPPINGS arxiv:math/000324v [math.fa] 2 Mar 2000 MANUEL GONZÁLEZ AND JOAQUÍN M. GUTIÉRREZ Abstract. We characterize the holomorphic mappings f between complex Banach

More information

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3 Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,

More information

Multiplication Operators with Closed Range in Operator Algebras

Multiplication Operators with Closed Range in Operator Algebras J. Ana. Num. Theor. 1, No. 1, 1-5 (2013) 1 Journal of Analysis & Number Theory An International Journal Multiplication Operators with Closed Range in Operator Algebras P. Sam Johnson Department of Mathematical

More information

(c) For each α R \ {0}, the mapping x αx is a homeomorphism of X.

(c) For each α R \ {0}, the mapping x αx is a homeomorphism of X. A short account of topological vector spaces Normed spaces, and especially Banach spaces, are basic ambient spaces in Infinite- Dimensional Analysis. However, there are situations in which it is necessary

More information

On metric characterizations of some classes of Banach spaces

On metric characterizations of some classes of Banach spaces On metric characterizations of some classes of Banach spaces Mikhail I. Ostrovskii January 12, 2011 Abstract. The first part of the paper is devoted to metric characterizations of Banach spaces with no

More information

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9 MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended

More information

Yuqing Chen, Yeol Je Cho, and Li Yang

Yuqing Chen, Yeol Je Cho, and Li Yang Bull. Korean Math. Soc. 39 (2002), No. 4, pp. 535 541 NOTE ON THE RESULTS WITH LOWER SEMI-CONTINUITY Yuqing Chen, Yeol Je Cho, and Li Yang Abstract. In this paper, we introduce the concept of lower semicontinuous

More information

Weak-Star Convergence of Convex Sets

Weak-Star Convergence of Convex Sets Journal of Convex Analysis Volume 13 (2006), No. 3+4, 711 719 Weak-Star Convergence of Convex Sets S. Fitzpatrick A. S. Lewis ORIE, Cornell University, Ithaca, NY 14853, USA aslewis@orie.cornell.edu www.orie.cornell.edu/

More information

Differentiability of Convex Functions on a Banach Space with Smooth Bump Function 1

Differentiability of Convex Functions on a Banach Space with Smooth Bump Function 1 Journal of Convex Analysis Volume 1 (1994), No.1, 47 60 Differentiability of Convex Functions on a Banach Space with Smooth Bump Function 1 Li Yongxin, Shi Shuzhong Nankai Institute of Mathematics Tianjin,

More information

Functional Analysis. Martin Brokate. 1 Normed Spaces 2. 2 Hilbert Spaces The Principle of Uniform Boundedness 32

Functional Analysis. Martin Brokate. 1 Normed Spaces 2. 2 Hilbert Spaces The Principle of Uniform Boundedness 32 Functional Analysis Martin Brokate Contents 1 Normed Spaces 2 2 Hilbert Spaces 2 3 The Principle of Uniform Boundedness 32 4 Extension, Reflexivity, Separation 37 5 Compact subsets of C and L p 46 6 Weak

More information

MATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES

MATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES MATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES PETE L. CLARK 4. Metric Spaces (no more lulz) Directions: This week, please solve any seven problems. Next week, please solve seven more. Starred parts of

More information

MA651 Topology. Lecture 10. Metric Spaces.

MA651 Topology. Lecture 10. Metric Spaces. MA65 Topology. Lecture 0. Metric Spaces. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Linear Algebra and Analysis by Marc Zamansky

More information

REPRESENTABLE BANACH SPACES AND UNIFORMLY GÂTEAUX-SMOOTH NORMS. Julien Frontisi

REPRESENTABLE BANACH SPACES AND UNIFORMLY GÂTEAUX-SMOOTH NORMS. Julien Frontisi Serdica Math. J. 22 (1996), 33-38 REPRESENTABLE BANACH SPACES AND UNIFORMLY GÂTEAUX-SMOOTH NORMS Julien Frontisi Communicated by G. Godefroy Abstract. It is proved that a representable non-separable Banach

More information

NOWHERE LOCALLY UNIFORMLY CONTINUOUS FUNCTIONS

NOWHERE LOCALLY UNIFORMLY CONTINUOUS FUNCTIONS NOWHERE LOCALLY UNIFORMLY CONTINUOUS FUNCTIONS K. Jarosz Southern Illinois University at Edwardsville, IL 606, and Bowling Green State University, OH 43403 kjarosz@siue.edu September, 995 Abstract. Suppose

More information

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................

More information

ON A HYBRID PROXIMAL POINT ALGORITHM IN BANACH SPACES

ON A HYBRID PROXIMAL POINT ALGORITHM IN BANACH SPACES U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 3, 2018 ISSN 1223-7027 ON A HYBRID PROXIMAL POINT ALGORITHM IN BANACH SPACES Vahid Dadashi 1 In this paper, we introduce a hybrid projection algorithm for a countable

More information

Division of the Humanities and Social Sciences. Supergradients. KC Border Fall 2001 v ::15.45

Division of the Humanities and Social Sciences. Supergradients. KC Border Fall 2001 v ::15.45 Division of the Humanities and Social Sciences Supergradients KC Border Fall 2001 1 The supergradient of a concave function There is a useful way to characterize the concavity of differentiable functions.

More information

The Hilbert Transform and Fine Continuity

The Hilbert Transform and Fine Continuity Irish Math. Soc. Bulletin 58 (2006), 8 9 8 The Hilbert Transform and Fine Continuity J. B. TWOMEY Abstract. It is shown that the Hilbert transform of a function having bounded variation in a finite interval

More information

HYERS-ULAM-RASSIAS STABILITY OF JENSEN S EQUATION AND ITS APPLICATION

HYERS-ULAM-RASSIAS STABILITY OF JENSEN S EQUATION AND ITS APPLICATION PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 16, Number 11, November 1998, Pages 3137 3143 S 000-9939(9804680- HYERS-ULAM-RASSIAS STABILITY OF JENSEN S EQUATION AND ITS APPLICATION SOON-MO JUNG

More information

David Hilbert was old and partly deaf in the nineteen thirties. Yet being a diligent

David Hilbert was old and partly deaf in the nineteen thirties. Yet being a diligent Chapter 5 ddddd dddddd dddddddd ddddddd dddddddd ddddddd Hilbert Space The Euclidean norm is special among all norms defined in R n for being induced by the Euclidean inner product (the dot product). A

More information

Tools from Lebesgue integration

Tools from Lebesgue integration Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given

More information

MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then

MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever

More information

SOME BANACH SPACE GEOMETRY

SOME BANACH SPACE GEOMETRY SOME BANACH SPACE GEOMETRY SVANTE JANSON 1. Introduction I have collected some standard facts about Banach spaces from various sources, see the references below for further results. Proofs are only given

More information

Peak Point Theorems for Uniform Algebras on Smooth Manifolds

Peak Point Theorems for Uniform Algebras on Smooth Manifolds Peak Point Theorems for Uniform Algebras on Smooth Manifolds John T. Anderson and Alexander J. Izzo Abstract: It was once conjectured that if A is a uniform algebra on its maximal ideal space X, and if

More information

I teach myself... Hilbert spaces

I teach myself... Hilbert spaces I teach myself... Hilbert spaces by F.J.Sayas, for MATH 806 November 4, 2015 This document will be growing with the semester. Every in red is for you to justify. Even if we start with the basic definition

More information

GENERALIZED POPOVICIU FUNCTIONAL EQUATIONS IN BANACH MODULES OVER A C ALGEBRA AND APPROXIMATE ALGEBRA HOMOMORPHISMS. Chun Gil Park

GENERALIZED POPOVICIU FUNCTIONAL EQUATIONS IN BANACH MODULES OVER A C ALGEBRA AND APPROXIMATE ALGEBRA HOMOMORPHISMS. Chun Gil Park NEW ZEALAND JOURNAL OF MATHEMATICS Volume 3 (003), 183 193 GENERALIZED POPOVICIU FUNCTIONAL EQUATIONS IN BANACH MODULES OVER A C ALGEBRA AND APPROXIMATE ALGEBRA HOMOMORPHISMS Chun Gil Park (Received March

More information

Compact operators on Banach spaces

Compact operators on Banach spaces Compact operators on Banach spaces Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto November 12, 2017 1 Introduction In this note I prove several things about compact

More information

Notions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy

Notions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.

More information

Introduction to Functional Analysis

Introduction to Functional Analysis Introduction to Functional Analysis Carnegie Mellon University, 21-640, Spring 2014 Acknowledgements These notes are based on the lecture course given by Irene Fonseca but may differ from the exact lecture

More information

7 Complete metric spaces and function spaces

7 Complete metric spaces and function spaces 7 Complete metric spaces and function spaces 7.1 Completeness Let (X, d) be a metric space. Definition 7.1. A sequence (x n ) n N in X is a Cauchy sequence if for any ɛ > 0, there is N N such that n, m

More information

REAL RENORMINGS ON COMPLEX BANACH SPACES

REAL RENORMINGS ON COMPLEX BANACH SPACES REAL RENORMINGS ON COMPLEX BANACH SPACES F. J. GARCÍA PACHECO AND A. MIRALLES Abstract. In this paper we provide two ways of obtaining real Banach spaces that cannot come from complex spaces. In concrete

More information

THE INVERSE FUNCTION THEOREM FOR LIPSCHITZ MAPS

THE INVERSE FUNCTION THEOREM FOR LIPSCHITZ MAPS THE INVERSE FUNCTION THEOREM FOR LIPSCHITZ MAPS RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA HOWARD@MATH.SC.EDU Abstract. This is an edited version of a

More information

AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES

AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES DUSTIN HEDMARK Abstract. A study of the conditions under which a topological space is metrizable, concluding with a proof of the Nagata Smirnov

More information

SOME REMARKS ON THE SPACE OF DIFFERENCES OF SUBLINEAR FUNCTIONS

SOME REMARKS ON THE SPACE OF DIFFERENCES OF SUBLINEAR FUNCTIONS APPLICATIONES MATHEMATICAE 22,3 (1994), pp. 419 426 S. G. BARTELS and D. PALLASCHKE (Karlsruhe) SOME REMARKS ON THE SPACE OF DIFFERENCES OF SUBLINEAR FUNCTIONS Abstract. Two properties concerning the space

More information

MA651 Topology. Lecture 9. Compactness 2.

MA651 Topology. Lecture 9. Compactness 2. MA651 Topology. Lecture 9. Compactness 2. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology

More information

Metric Spaces Lecture 17

Metric Spaces Lecture 17 Metric Spaces Lecture 17 Homeomorphisms At the end of last lecture an example was given of a bijective continuous function f such that f 1 is not continuous. For another example, consider the sets T =

More information

("-1/' .. f/ L) I LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERA TORS. R. T. Rockafellar. MICHIGAN MATHEMATICAL vol. 16 (1969) pp.

(-1/' .. f/ L) I LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERA TORS. R. T. Rockafellar. MICHIGAN MATHEMATICAL vol. 16 (1969) pp. I l ("-1/'.. f/ L) I LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERA TORS R. T. Rockafellar from the MICHIGAN MATHEMATICAL vol. 16 (1969) pp. 397-407 JOURNAL LOCAL BOUNDEDNESS OF NONLINEAR, MONOTONE OPERATORS

More information

On constraint qualifications with generalized convexity and optimality conditions

On constraint qualifications with generalized convexity and optimality conditions On constraint qualifications with generalized convexity and optimality conditions Manh-Hung Nguyen, Do Van Luu To cite this version: Manh-Hung Nguyen, Do Van Luu. On constraint qualifications with generalized

More information

Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011

Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011 Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011 Section 2.6 (cont.) Properties of Real Functions Here we first study properties of functions from R to R, making use of the additional structure

More information

FRAGMENTABILITY OF ROTUND BANACH SPACES. Scott Sciffer. lim <Jl(x+A.y)- $(x) A-tO A

FRAGMENTABILITY OF ROTUND BANACH SPACES. Scott Sciffer. lim <Jl(x+A.y)- $(x) A-tO A 222 FRAGMENTABILITY OF ROTUND BANACH SPACES Scott Sciffer Introduction A topological. space X is said to be fragmented by a metric p if for every e > 0 and every subset Y of X there exists a nonempty relatively

More information

Brøndsted-Rockafellar property of subdifferentials of prox-bounded functions. Marc Lassonde Université des Antilles et de la Guyane

Brøndsted-Rockafellar property of subdifferentials of prox-bounded functions. Marc Lassonde Université des Antilles et de la Guyane Conference ADGO 2013 October 16, 2013 Brøndsted-Rockafellar property of subdifferentials of prox-bounded functions Marc Lassonde Université des Antilles et de la Guyane Playa Blanca, Tongoy, Chile SUBDIFFERENTIAL

More information

ON THE CONVERGENCE OF GREEDY ALGORITHMS FOR INITIAL SEGMENTS OF THE HAAR BASIS

ON THE CONVERGENCE OF GREEDY ALGORITHMS FOR INITIAL SEGMENTS OF THE HAAR BASIS ON THE CONVERGENCE OF GREEDY ALGORITHMS FOR INITIAL SEGMENTS OF THE HAAR BASIS S. J. DILWORTH, E. ODELL, TH. SCHLUMPRECHT, AND ANDRÁS ZSÁK Abstract. We consider the X-Greedy Algorithm and the Dual Greedy

More information

Chapter 2 Metric Spaces

Chapter 2 Metric Spaces Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics

More information

MAXIMALITY OF SUMS OF TWO MAXIMAL MONOTONE OPERATORS

MAXIMALITY OF SUMS OF TWO MAXIMAL MONOTONE OPERATORS MAXIMALITY OF SUMS OF TWO MAXIMAL MONOTONE OPERATORS JONATHAN M. BORWEIN, FRSC Abstract. We use methods from convex analysis convex, relying on an ingenious function of Simon Fitzpatrick, to prove maximality

More information

Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm

Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm Chapter 13 Radon Measures Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm (13.1) f = sup x X f(x). We want to identify

More information

arxiv: v1 [math.oc] 21 Mar 2015

arxiv: v1 [math.oc] 21 Mar 2015 Convex KKM maps, monotone operators and Minty variational inequalities arxiv:1503.06363v1 [math.oc] 21 Mar 2015 Marc Lassonde Université des Antilles, 97159 Pointe à Pitre, France E-mail: marc.lassonde@univ-ag.fr

More information

The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space

The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space Vol. 45 No. 4 SCIENCE IN CHINA Series A April 2002 The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space DING Guanggui Department

More information

FUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES. Yılmaz Yılmaz

FUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES. Yılmaz Yılmaz Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 33, 2009, 335 353 FUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES Yılmaz Yılmaz Abstract. Our main interest in this

More information

Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem

Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem 56 Chapter 7 Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem Recall that C(X) is not a normed linear space when X is not compact. On the other hand we could use semi

More information

PROBLEMS. (b) (Polarization Identity) Show that in any inner product space

PROBLEMS. (b) (Polarization Identity) Show that in any inner product space 1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization

More information

Aliprantis, Border: Infinite-dimensional Analysis A Hitchhiker s Guide

Aliprantis, Border: Infinite-dimensional Analysis A Hitchhiker s Guide aliprantis.tex May 10, 2011 Aliprantis, Border: Infinite-dimensional Analysis A Hitchhiker s Guide Notes from [AB2]. 1 Odds and Ends 2 Topology 2.1 Topological spaces Example. (2.2) A semimetric = triangle

More information

Normed Vector Spaces and Double Duals

Normed Vector Spaces and Double Duals Normed Vector Spaces and Double Duals Mathematics 481/525 In this note we look at a number of infinite-dimensional R-vector spaces that arise in analysis, and we consider their dual and double dual spaces

More information

YET MORE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS

YET MORE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 103, Number 3, July 1988 YET MORE ON THE DIFFERENTIABILITY OF CONVEX FUNCTIONS JOHN RAINWATER (Communicated by William J. Davis) ABSTRACT. Generic

More information

Metric Space Topology (Spring 2016) Selected Homework Solutions. HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y)

Metric Space Topology (Spring 2016) Selected Homework Solutions. HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y) Metric Space Topology (Spring 2016) Selected Homework Solutions HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y) d(z, w) d(x, z) + d(y, w) holds for all w, x, y, z X.

More information

Course 212: Academic Year Section 1: Metric Spaces

Course 212: Academic Year Section 1: Metric Spaces Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........

More information

Introduction to Real Analysis Alternative Chapter 1

Introduction to Real Analysis Alternative Chapter 1 Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces

More information

Your first day at work MATH 806 (Fall 2015)

Your first day at work MATH 806 (Fall 2015) Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies

More information

B. Appendix B. Topological vector spaces

B. Appendix B. Topological vector spaces B.1 B. Appendix B. Topological vector spaces B.1. Fréchet spaces. In this appendix we go through the definition of Fréchet spaces and their inductive limits, such as they are used for definitions of function

More information

08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms

08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms (February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops

More information

A NOTE ON LINEAR FUNCTIONAL NORMS

A NOTE ON LINEAR FUNCTIONAL NORMS A NOTE ON LINEAR FUNCTIONAL NORMS YIFEI PAN AND MEI WANG Abstract. For a vector u in a normed linear space, Hahn-Banach Theorem provides the existence of a linear functional f, f(u) = u such that f = 1.

More information

THEOREMS, ETC., FOR MATH 516

THEOREMS, ETC., FOR MATH 516 THEOREMS, ETC., FOR MATH 516 Results labeled Theorem Ea.b.c (or Proposition Ea.b.c, etc.) refer to Theorem c from section a.b of Evans book (Partial Differential Equations). Proposition 1 (=Proposition

More information

arxiv: v1 [math.fa] 11 Oct 2018

arxiv: v1 [math.fa] 11 Oct 2018 SOME REMARKS ON BIRKHOFF-JAMES ORTHOGONALITY OF LINEAR OPERATORS arxiv:1810.04845v1 [math.fa] 11 Oct 2018 DEBMALYA SAIN, KALLOL PAUL AND ARPITA MAL Abstract. We study Birkhoff-James orthogonality of compact

More information

A PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE. Myung-Hyun Cho and Jun-Hui Kim. 1. Introduction

A PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE. Myung-Hyun Cho and Jun-Hui Kim. 1. Introduction Comm. Korean Math. Soc. 16 (2001), No. 2, pp. 277 285 A PROOF OF A CONVEX-VALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE Myung-Hyun Cho and Jun-Hui Kim Abstract. The purpose of this paper

More information

Chapter 3: Baire category and open mapping theorems

Chapter 3: Baire category and open mapping theorems MA3421 2016 17 Chapter 3: Baire category and open mapping theorems A number of the major results rely on completeness via the Baire category theorem. 3.1 The Baire category theorem 3.1.1 Definition. A

More information

Math 209B Homework 2

Math 209B Homework 2 Math 29B Homework 2 Edward Burkard Note: All vector spaces are over the field F = R or C 4.6. Two Compactness Theorems. 4. Point Set Topology Exercise 6 The product of countably many sequentally compact

More information

Examples of Convex Functions and Classifications of Normed Spaces

Examples of Convex Functions and Classifications of Normed Spaces Journal of Convex Analysis Volume 1 (1994), No.1, 61 73 Examples of Convex Functions and Classifications of Normed Spaces Jon Borwein 1 Department of Mathematics and Statistics, Simon Fraser University

More information

POINTWISE PRODUCTS OF UNIFORMLY CONTINUOUS FUNCTIONS

POINTWISE PRODUCTS OF UNIFORMLY CONTINUOUS FUNCTIONS SARAJEVO JOURNAL OF MATHEMATICS Vol.1 (13) (2005), 117 127 POINTWISE PRODUCTS OF UNIFORMLY CONTINUOUS FUNCTIONS SAM B. NADLER, JR. Abstract. The problem of characterizing the metric spaces on which the

More information

LECTURE 15: COMPLETENESS AND CONVEXITY

LECTURE 15: COMPLETENESS AND CONVEXITY LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other

More information

A criterion of Γ -nullness and differentiability of convex and quasiconvex functions

A criterion of Γ -nullness and differentiability of convex and quasiconvex functions STUDIA MATHEMATICA 227 (2) (2015) A criterion of Γ -nullness and differentiability of convex and quasiconvex functions by Jaroslav Tišer and Luděk Zajíček (Praha) Abstract. We introduce a criterion for

More information

ON SPACES IN WHICH COMPACT-LIKE SETS ARE CLOSED, AND RELATED SPACES

ON SPACES IN WHICH COMPACT-LIKE SETS ARE CLOSED, AND RELATED SPACES Commun. Korean Math. Soc. 22 (2007), No. 2, pp. 297 303 ON SPACES IN WHICH COMPACT-LIKE SETS ARE CLOSED, AND RELATED SPACES Woo Chorl Hong Reprinted from the Communications of the Korean Mathematical Society

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: WEAK AND WEAK* CONVERGENCE

FUNCTIONAL ANALYSIS LECTURE NOTES: WEAK AND WEAK* CONVERGENCE FUNCTIONAL ANALYSIS LECTURE NOTES: WEAK AND WEAK* CONVERGENCE CHRISTOPHER HEIL 1. Weak and Weak* Convergence of Vectors Definition 1.1. Let X be a normed linear space, and let x n, x X. a. We say that

More information

arxiv:math/ v1 [math.fa] 26 Oct 1993

arxiv:math/ v1 [math.fa] 26 Oct 1993 arxiv:math/9310217v1 [math.fa] 26 Oct 1993 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M.I.Ostrovskii Abstract. It is proved that there exist complemented subspaces of countable topological

More information

ON GAP FUNCTIONS OF VARIATIONAL INEQUALITY IN A BANACH SPACE. Sangho Kum and Gue Myung Lee. 1. Introduction

ON GAP FUNCTIONS OF VARIATIONAL INEQUALITY IN A BANACH SPACE. Sangho Kum and Gue Myung Lee. 1. Introduction J. Korean Math. Soc. 38 (2001), No. 3, pp. 683 695 ON GAP FUNCTIONS OF VARIATIONAL INEQUALITY IN A BANACH SPACE Sangho Kum and Gue Myung Lee Abstract. In this paper we are concerned with theoretical properties

More information

The Measure Problem. Louis de Branges Department of Mathematics Purdue University West Lafayette, IN , USA

The Measure Problem. Louis de Branges Department of Mathematics Purdue University West Lafayette, IN , USA The Measure Problem Louis de Branges Department of Mathematics Purdue University West Lafayette, IN 47907-2067, USA A problem of Banach is to determine the structure of a nonnegative (countably additive)

More information

Functional Analysis Exercise Class

Functional Analysis Exercise Class Functional Analysis Exercise Class Week 9 November 13 November Deadline to hand in the homeworks: your exercise class on week 16 November 20 November Exercises (1) Show that if T B(X, Y ) and S B(Y, Z)

More information

HERMITIAN OPERATORS ON BANACH ALGEBRAS OF VECTOR-VALUED LIPSCHITZ MAPS

HERMITIAN OPERATORS ON BANACH ALGEBRAS OF VECTOR-VALUED LIPSCHITZ MAPS HERMITIAN OPERATORS ON BANACH ALGEBRAS OF VECTOR-VALUED LIPSCHITZ MAPS (JOINT WORK WITH OSAMU HATORI) SHIHO OI NIIGATA PREFECTURAL NAGAOKA HIGH SCHOOL Abstract. Let H be a complex Hilbert space and [,

More information

HILBERT SPACES AND THE RADON-NIKODYM THEOREM. where the bar in the first equation denotes complex conjugation. In either case, for any x V define

HILBERT SPACES AND THE RADON-NIKODYM THEOREM. where the bar in the first equation denotes complex conjugation. In either case, for any x V define HILBERT SPACES AND THE RADON-NIKODYM THEOREM STEVEN P. LALLEY 1. DEFINITIONS Definition 1. A real inner product space is a real vector space V together with a symmetric, bilinear, positive-definite mapping,

More information

Filters in Analysis and Topology

Filters in Analysis and Topology Filters in Analysis and Topology David MacIver July 1, 2004 Abstract The study of filters is a very natural way to talk about convergence in an arbitrary topological space, and carries over nicely into

More information