DOCSLIB.ORG

FUNCTIONAL ANALYSIS 1. Banach and Hilbert Spaces in What

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
FUNCTIONAL ANALYSIS 1. Banach and Hilbert Spaces in What FUNCTIONAL ANALYSIS PIOTR HAJLASZ 1. Banach and Hilbert spaces In what follows K will denote R of C. Definition. A normed space is a pair (X, k · k), where X is a linear space over K and k · k : X → [0, ∞) is a function, called a norm, such that (1) kx + yk ≤ kxk + kyk for all x, y ∈ X; (2) kαxk = |α|kxk for all x ∈ X and α ∈ K; (3) kxk = 0 if and only if x = 0. Since kx − yk ≤ kx − zk + kz − yk for all x, y, z ∈ X, d(x, y) = kx − yk defines a metric in a normed space. In what follows normed paces will always be regarded as metric spaces with respect to the metric d. A normed space is called a Banach space if it is complete with respect to the metric d. Definition. Let X be a linear space over K (=R or C). The inner product (scalar product) is a function h·, ·i : X × X → K such that (1) hx, xi ≥ 0; (2) hx, xi = 0 if and only if x = 0; (3) hαx, yi = αhx, yi; (4) hx1 + x2, yi = hx1, yi + hx2, yi; (5) hx, yi = hy, xi, for all x, x1, x2, y ∈ X and all α ∈ K. As an obvious corollary we obtain hx, y1 + y2i = hx, y1i + hx, y2i, hx, αyi = αhx, yi , Date: February 12, 2009. 1 2 PIOTR HAJLASZ for all x, y1, y2 ∈ X and α ∈ K. For a space with an inner product we define kxk = phx, xi . Lemma 1.1 (Schwarz inequality). If X is a space with an inner product h·, ·i, then |hx, yi| ≤ kxk kyk for all x, y ∈ X. Proof. We can assume that hx, yi 6= 0. For x, y ∈ X and t ∈ R we have 0 ≤ hx + ty, x + tyi = hx, xi + t[hx, yi + hy, xi] + t2hy, yi = hx, xi + 2t re hx, yi + t2hy, yi . We obtained a quadratic function of a variable t which is nonnegative and hence 0 ≥ ∆ = 4(re hx, yi)2 − 4hx, xihy, yi , (re hx, yi)2 ≤ hx, xihy, yi . If |α| = 1, then replacing y by αy we obtain hx, xihy, yi = hx, xihαy, αyi ≥ (re hx, αyi)2 = (re (αhx, yi))2 . In particular for α = hx, yi/|hx, yi| we have !!2 hx, yi hx, xihy, yi ≥ re hx, yi = |hx, yi|2 . |hx, yi| This completes the proof. 2 Corollary 1.2. If X is a space with an inner product h·, ·i, then kxk = phx, yi is a norm. Proof. The properties kαxk = |α|kxk and kxk = 0 if and only if x = 0 are obvious. To prove the last property we need to apply the Schwarz inequality. kx + yk2 = hx + y, x + yi = hx, xi + 2 re hx, yi + hy, yi ≤ kxk2 + 2kxk kyk + kyk2 = (kxk + kyk)2 . Definition. A space with an inner product h·, ·i is called a Hilbert space if it is a Banach space with respect to the norm kxk = phx, xi . Proposition 1.3 (The Polarization Identity). Let h·, ·i be an inner product in X. FUNCTIONAL ANALYSIS 3 (1) If K = R, then 1 hx, yi = (kx + yk2 − kx − yk2) , 4 for all x, y ∈ X. (2) If K = C, then 1 1 hx, yi = kx + yk2 − kx − yk2 − i kix + yk2 − kix − yk2 4 4 for all x, y ∈ X. Proof is left as an easy exercise. Theorem 1.4. Let (X, k · k) be a normed space over K (= R or C). Then there is an inner product h·, ·i such that kxk = phx, xi if and only if the norm satisfies the Parallelogram Law, i.e. kx + yk2 + kx − yk2 = 2kxk2 + 2kyk2 for all x, y ∈ X . The Parallelogram Law has a nice geometric interpretation. Proof. The implication ⇒ follows from a direct computation. To prove the other implication ⇐ we define the inner product using the Polarization Iden- tities and we check that it has all the required properties. We leave the details as an exercise. 2 n 1.1. Examples. 1. C with respect to each of the following norms is a Banach space n X kxk1 = |xi|, i=1 kxk∞ = max |xi|, i=1,2,...,n n !1/p X p kxkp = |xi| , 1 ≤ p < ∞ , i=1 n n where x = (x1, . , xn) ∈ C . The Banach space (C , k · kp) is denoted by p `n. 4 PIOTR HAJLASZ n 2. C with the inner product n X hx, yi = xiyi i=1 is a Hilbert space. Note that the corresponding norm is n !1/2 p X 2 hx, xi = |xi| = kxk2 i=1 2 so `n is a Hilbert space. The metric associated with the norm is v u n uX 2 d(x, y) = t |xi − yi| , i=1 i.e. it is the Euclidean metric. p 3. The spaces `n were defined over K = C, but we can also do the same n n construction for K = R by replacing C by R . The resulting space is also p denoted by `n, but in each situation it will be clear whether we talk about p the real or complex space `n so there is no danger of a confusion. p 4. Consider `2 over K = R. Then the shape of the unit ball {x : kxk ≤ 1} is n 5. A subset of the Euclidean space R is called an ellipsoid if it is the image n n n of the unit ball in R under a nondegenerate linear mapping L : R → R (i.e. det L 6= 0). n n For every ellipsoid E in R there is an inner product in R such that n n E is the unit ball in the associated norm. Indeed, if L : R → R is an isomorphism such that E = L(Bn(0, 1)), then it suffices to define the inner product as hx, yi = (L−1x) · (L−1y) n where · stands for the standard inner product in R . n n More precisely, if L : R → R is non-degenerate, then according to the polar decomposition theorem (P. Lax, Linear Algebra, p. 139) L = RU FUNCTIONAL ANALYSIS 5 where U is unitary and R is positive self-adjoint. The mapping R can be computed explicitly √ LLT = RUU T RT = R2,R = LLT . According to the spectral theorem there is an orthonormal basis v1, . vn in n R (with respect to the standard inner product) such that λ1 0 λ2 R = .. . 0 λn in this basis. That means the mapping L has the following structure. First we rotate (the mapping U) and then we apply R which has a simple geometric meaning of extending the length of vectors v1, . , vn by factors λ1, . , λn. Now if Bn(0, 1) is the unit ball, then U(Bn(0, 1)) is also the unit ball, so n n L(B (0, 1)) = R(B (0, 1)) is an ellipsoid√ with semi-axes λ1v1, . , λnvn of the lengths being eigenvalues of R = LLT . These numbers are called sin- gular values of L. Now the ellipsoid E = L(Bn(0, 1)) is the unit ball for the inner product n n n X X X aibi a v , b v = . i i i i λ2 i=1 i=1 i=1 i We will see later (Corollary 5.12) any real inner product space space H 2 n of dimension n is isometrically isomorphic to `n, i.e. R with the standard 2 inner product, so if L : `n → H is this isometric isomorphism, the unit ball in H is L(Bn(0, 1)), so it is an ellipsoid. Thus we proved. n Theorem 1.5. A convex set in R is a unit ball for a norm associated with an inner product if and only if it is an ellipsoid. ∞ ∞ 6. ` , the space of all bounded (complex, real) sequences x = (an)n=1 with the norm kxk∞ = sup |xn| n is a Banach space. This is very easy to check. 6 PIOTR HAJLASZ 7. c1, the space of all (complex, real) convergent sequences with the norm k · k∞ is a Banach space. 8. c0, the space of all (complex, real) sequences that converge to zero with the norm k · k∞ is a Banach space. ∞ 9. Note that c0 ⊂ c ⊂ ` and both c0 and c are closed linear subspaces of `∞ with respect to the metric generated by the norm. ∞ Exercise. Prove that ` , c and c0 are Banach spaces. ∞ Exercise. Prove that the spaces c and c0 are separable, while ` is not. p ∞ 10. ` , 1 ≤ p < ∞ is the space of all (complex, real) sequences x = (xn)n=1 such that ∞ !1/p X p kxkp = |xn| . n=1 It follows from the Minkowski inequality for sequences that k · kp is a norm and that `p is a linear space. We will prove now that `p is a Banach space, n ∞ p i.e. that it is complete. Let xn = (ai )i=1 be a Cauchy sequence in ` , i.e. for every ε > 0 there is N such that for all n, m > N ∞ !1/p X n m p kxn − xmkp = |ai − ai | < ε . i=1 n ∞ Hence for each i the sequence (ai )n=1 is a Cauchy sequence in K (= C or n R). Let ai = limn→∞ ai . Fix an integer k. Then for n, m > N we have k X n m p p |ai − ai | < ε i=1 and passing to the limit as m → ∞ yields k X n p p |ai − ai| ≤ ε . i=1 Now taking the limit as k → ∞ we obtain ∞ X n p p |ai − ai| ≤ ε , i=1 i.e.
Load more

Recommended publications
  • On Quasi Norm Attaining Operators Between Banach Spaces
    ON QUASI NORM ATTAINING OPERATORS BETWEEN BANACH SPACES GEUNSU CHOI, YUN SUNG CHOI, MINGU JUNG, AND MIGUEL MART´IN Abstract. We provide a characterization of the Radon-Nikod´ymproperty in terms of the denseness of bounded linear operators which attain their norm in a weak sense, which complement the one given by Bourgain and Huff in the 1970's. To this end, we introduce the following notion: an operator T : X ÝÑ Y between the Banach spaces X and Y is quasi norm attaining if there is a sequence pxnq of norm one elements in X such that pT xnq converges to some u P Y with }u}“}T }. Norm attaining operators in the usual (or strong) sense (i.e. operators for which there is a point in the unit ball where the norm of its image equals the norm of the operator) and also compact operators satisfy this definition. We prove that strong Radon-Nikod´ymoperators can be approximated by quasi norm attaining operators, a result which does not hold for norm attaining operators in the strong sense. This shows that this new notion of quasi norm attainment allows to characterize the Radon-Nikod´ymproperty in terms of denseness of quasi norm attaining operators for both domain and range spaces, completing thus a characterization by Bourgain and Huff in terms of norm attaining operators which is only valid for domain spaces and it is actually false for range spaces (due to a celebrated example by Gowers of 1990). A number of other related results are also included in the paper: we give some positive results on the denseness of norm attaining Lipschitz maps, norm attaining multilinear maps and norm attaining polynomials, characterize both finite dimensionality and reflexivity in terms of quasi norm attaining operators, discuss conditions to obtain that quasi norm attaining operators are actually norm attaining, study the relationship with the norm attainment of the adjoint operator and, finally, present some stability results.
    [Show full text]
  • Fundamentals of Functional Analysis Kluwer Texts in the Mathematical Sciences
    Fundamentals of Functional Analysis Kluwer Texts in the Mathematical Sciences VOLUME 12 A Graduate-Level Book Series The titles published in this series are listed at the end of this volume. Fundamentals of Functional Analysis by S. S. Kutateladze Sobolev Institute ofMathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia Springer-Science+Business Media, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress ISBN 978-90-481-4661-1 ISBN 978-94-015-8755-6 (eBook) DOI 10.1007/978-94-015-8755-6 Translated from OCHOBbI Ij)YHK~HOHaJIl>HODO aHaJIHsa. J/IS;l\~ 2, ;l\OIIOJIHeHHoe., Sobo1ev Institute of Mathematics, Novosibirsk, © 1995 S. S. Kutate1adze Printed on acid-free paper All Rights Reserved © 1996 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1996. Softcover reprint of the hardcover 1st edition 1996 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Contents Preface to the English Translation ix Preface to the First Russian Edition x Preface to the Second Russian Edition xii Chapter 1. An Excursion into Set Theory 1.1. Correspondences . 1 1.2. Ordered Sets . 3 1.3. Filters . 6 Exercises . 8 Chapter 2. Vector Spaces 2.1. Spaces and Subspaces ... ......................... 10 2.2. Linear Operators . 12 2.3. Equations in Operators ........................ .. 15 Exercises . 18 Chapter 3. Convex Analysis 3.1.
    [Show full text]
  • Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces
    Axioms 2013, 2, 224-270; doi:10.3390/axioms2020224 OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Article Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces Sergey Ajiev School of Mathematical Sciences, Faculty of Science, University of Technology-Sydney, P.O. Box 123, Broadway, NSW 2007, Australia; E-Mail: [email protected] Received: 4 March 2013; in revised form: 12 May 2013 / Accepted: 14 May 2013 / Published: 23 May 2013 Abstract: We introduce and study Clarkson, Dol’nikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Schaffer¨ constants in order to establish quantitative versions of Hahn-Banach separability theorem and to characterise the isometric extendability of Holder-Lipschitz¨ mappings. Abstract results are further applied to the spaces and pairs from the wide classes IG and IG+ and non-commutative Lp-spaces. The intimate relation between the subspaces and quotients of the IG-spaces on one side and various types of anisotropic Besov, Lizorkin-Triebel and Sobolev spaces of functions on open subsets of an Euclidean space defined in terms of differences, local polynomial approximations, wavelet decompositions and other means (as well as the duals and the lp-sums of all these spaces) on the other side, allows us to present the algorithm of extending the main results of the article to the latter spaces and pairs. Special attention is paid to the matter of sharpness. Our approach is quasi-Euclidean in its nature because it relies on the extrapolation of properties of Hilbert spaces and the study of 1-complemented subspaces of the spaces under consideration.
    [Show full text]
  • Uniform Boundedness Principle for Unbounded Operators
    UNIFORM BOUNDEDNESS PRINCIPLE FOR UNBOUNDED OPERATORS C. GANESA MOORTHY and CT. RAMASAMY Abstract. A uniform boundedness principle for unbounded operators is derived. A particular case is: Suppose fTigi2I is a family of linear mappings of a Banach space X into a normed space Y such that fTix : i 2 Ig is bounded for each x 2 X; then there exists a dense subset A of the open unit ball in X such that fTix : i 2 I; x 2 Ag is bounded. A closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings as consequences of this principle. Some applications of this principle are also obtained. 1. Introduction There are many forms for uniform boundedness principle. There is no known evidence for this principle for unbounded operators which generalizes classical uniform boundedness principle for bounded operators. The second section presents a uniform boundedness principle for unbounded operators. An application to derive Hellinger-Toeplitz theorem is also obtained in this section. A JJ J I II closed graph theorem and a bounded inverse theorem are obtained for families of linear mappings in the third section as consequences of this principle. Go back Let us assume the following: Every vector space X is over R or C. An α-seminorm (0 < α ≤ 1) is a mapping p: X ! [0; 1) such that p(x + y) ≤ p(x) + p(y), p(ax) ≤ jajαp(x) for all x; y 2 X Full Screen Close Received November 7, 2013. 2010 Mathematics Subject Classification. Primary 46A32, 47L60. Key words and phrases.
    [Show full text]
  • Proximal Analysis and Boundaries of Closed Sets in Banach Space, Part I: Theory
    Can. J. Math., Vol. XXXVIII, No. 2, 1986, pp. 431-452 PROXIMAL ANALYSIS AND BOUNDARIES OF CLOSED SETS IN BANACH SPACE, PART I: THEORY J. M. BORWEIN AND H. M. STROJWAS 0. Introduction. As various types of tangent cones, generalized deriva­ tives and subgradients prove to be a useful tool in nonsmooth optimization and nonsmooth analysis, we witness a considerable interest in analysis of their properties, relations and applications. Recently, Treiman [18] proved that the Clarke tangent cone at a point to a closed subset of a Banach space contains the limit inferior of the contingent cones to the set at neighbouring points. We provide a considerable strengthening of this result for reflexive spaces. Exploring the analogous inclusion in which the contingent cones are replaced by pseudocontingent cones we have observed that it does not hold any longer in a general Banach space, however it does in reflexive spaces. Among the several basic relations we have discovered is the following one: the Clarke tangent cone at a point to a closed subset of a reflexive Banach space is equal to the limit inferior of the weak (pseudo) contingent cones to the set at neighbouring points. This generalizes the results of Penot [14] and Cornet [7] for finite dimensional spaces and of Penot [14] for reflexive spaces. What is more we show that this equality characterizes reflex­ ive spaces among Banach spaces, similarly as the analogous equality with the contingent cones instead of the weak contingent cones, characterizes finite dimensional spaces among Banach spaces. We also present several other related characterizations of reflexive spaces and variants of the considered inclusions for boundedly relatively weakly compact sets.
    [Show full text]
  • On the Boundary Between Mereology and Topology
    On the Boundary Between Mereology and Topology Achille C. Varzi Istituto per la Ricerca Scientifica e Tecnologica, I-38100 Trento, Italy [email protected] (Final version published in Roberto Casati, Barry Smith, and Graham White (eds.), Philosophy and the Cognitive Sciences. Proceedings of the 16th International Wittgenstein Symposium, Vienna, Hölder-Pichler-Tempsky, 1994, pp. 423–442.) 1. Introduction Much recent work aimed at providing a formal ontology for the common-sense world has emphasized the need for a mereological account to be supplemented with topological concepts and principles. There are at least two reasons under- lying this view. The first is truly metaphysical and relates to the task of charac- terizing individual integrity or organic unity: since the notion of connectedness runs afoul of plain mereology, a theory of parts and wholes really needs to in- corporate a topological machinery of some sort. The second reason has been stressed mainly in connection with applications to certain areas of artificial in- telligence, most notably naive physics and qualitative reasoning about space and time: here mereology proves useful to account for certain basic relation- ships among things or events; but one needs topology to account for the fact that, say, two events can be continuous with each other, or that something can be inside, outside, abutting, or surrounding something else. These motivations (at times combined with others, e.g., semantic transpar- ency or computational efficiency) have led to the development of theories in which both mereological and topological notions play a pivotal role. How ex- actly these notions are related, however, and how the underlying principles should interact with one another, is still a rather unexplored issue.
    [Show full text]
  • Functional Analysis Lecture Notes Chapter 2. Operators on Hilbert Spaces
    FUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 2. OPERATORS ON HILBERT SPACES CHRISTOPHER HEIL 1. Elementary Properties and Examples First recall the basic definitions regarding operators. Definition 1.1 (Continuous and Bounded Operators). Let X, Y be normed linear spaces, and let L: X Y be a linear operator. ! (a) L is continuous at a point f X if f f in X implies Lf Lf in Y . 2 n ! n ! (b) L is continuous if it is continuous at every point, i.e., if fn f in X implies Lfn Lf in Y for every f. ! ! (c) L is bounded if there exists a finite K 0 such that ≥ f X; Lf K f : 8 2 k k ≤ k k Note that Lf is the norm of Lf in Y , while f is the norm of f in X. k k k k (d) The operator norm of L is L = sup Lf : k k kfk=1 k k (e) We let (X; Y ) denote the set of all bounded linear operators mapping X into Y , i.e., B (X; Y ) = L: X Y : L is bounded and linear : B f ! g If X = Y = X then we write (X) = (X; X). B B (f) If Y = F then we say that L is a functional. The set of all bounded linear functionals on X is the dual space of X, and is denoted X0 = (X; F) = L: X F : L is bounded and linear : B f ! g We saw in Chapter 1 that, for a linear operator, boundedness and continuity are equivalent.
    [Show full text]
  • MAS4107 Linear Algebra 2 Linear Maps And
    Introduction Groups and Fields Vector Spaces Subspaces, Linear . Bases and Coordinates MAS4107 Linear Algebra 2 Linear Maps and . Change of Basis Peter Sin More on Linear Maps University of Florida Linear Endomorphisms email: [email protected]fl.edu Quotient Spaces Spaces of Linear . General Prerequisites Direct Sums Minimal polynomial Familiarity with the notion of mathematical proof and some experience in read- Bilinear Forms ing and writing proofs. Familiarity with standard mathematical notation such as Hermitian Forms summations and notations of set theory. Euclidean and . Self-Adjoint Linear . Linear Algebra Prerequisites Notation Familiarity with the notion of linear independence. Gaussian elimination (reduction by row operations) to solve systems of equations. This is the most important algorithm and it will be assumed and used freely in the classes, for example to find JJ J I II coordinate vectors with respect to basis and to compute the matrix of a linear map, to test for linear dependence, etc. The determinant of a square matrix by cofactors Back and also by row operations. Full Screen Close Quit Introduction 0. Introduction Groups and Fields Vector Spaces These notes include some topics from MAS4105, which you should have seen in one Subspaces, Linear . form or another, but probably presented in a totally different way. They have been Bases and Coordinates written in a terse style, so you should read very slowly and with patience. Please Linear Maps and . feel free to email me with any questions or comments. The notes are in electronic Change of Basis form so sections can be changed very easily to incorporate improvements.
    [Show full text]
  • On the Origin and Early History of Functional Analysis
    U.U.D.M. Project Report 2008:1 On the origin and early history of functional analysis Jens Lindström Examensarbete i matematik, 30 hp Handledare och examinator: Sten Kaijser Januari 2008 Department of Mathematics Uppsala University Abstract In this report we will study the origins and history of functional analysis up until 1918. We begin by studying ordinary and partial differential equations in the 18th and 19th century to see why there was a need to develop the concepts of functions and limits. We will see how a general theory of infinite systems of equations and determinants by Helge von Koch were used in Ivar Fredholm’s 1900 paper on the integral equation b Z ϕ(s) = f(s) + λ K(s, t)f(t)dt (1) a which resulted in a vast study of integral equations. One of the most enthusiastic followers of Fredholm and integral equation theory was David Hilbert, and we will see how he further developed the theory of integral equations and spectral theory. The concept introduced by Fredholm to study sets of transformations, or operators, made Maurice Fr´echet realize that the focus should be shifted from particular objects to sets of objects and the algebraic properties of these sets. This led him to introduce abstract spaces and we will see how he introduced the axioms that defines them. Finally, we will investigate how the Lebesgue theory of integration were used by Frigyes Riesz who was able to connect all theory of Fredholm, Fr´echet and Lebesgue to form a general theory, and a new discipline of mathematics, now known as functional analysis.
    [Show full text]
  • Using Functional Analysis and Sobolev Spaces to Solve Poisson’S Equation
    USING FUNCTIONAL ANALYSIS AND SOBOLEV SPACES TO SOLVE POISSON'S EQUATION YI WANG Abstract. We study Banach and Hilbert spaces with an eye to- wards defining weak solutions to elliptic PDE. Using Lax-Milgram we prove that weak solutions to Poisson's equation exist under certain conditions. Contents 1. Introduction 1 2. Banach spaces 2 3. Weak topology, weak star topology and reflexivity 6 4. Lower semicontinuity 11 5. Hilbert spaces 13 6. Sobolev spaces 19 References 21 1. Introduction We will discuss the following problem in this paper: let Ω be an open and connected subset in R and f be an L2 function on Ω, is there a solution to Poisson's equation (1) −∆u = f? From elementary partial differential equations class, we know if Ω = R, we can solve Poisson's equation using the fundamental solution to Laplace's equation. However, if we just take Ω to be an open and connected set, the above method is no longer useful. In addition, for arbitrary Ω and f, a C2 solution does not always exist. Therefore, instead of finding a strong solution, i.e., a C2 function which satisfies (1), we integrate (1) against a test function φ (a test function is a Date: September 28, 2016. 1 2 YI WANG smooth function compactly supported in Ω), integrate by parts, and arrive at the equation Z Z 1 (2) rurφ = fφ, 8φ 2 Cc (Ω): Ω Ω So intuitively we want to find a function which satisfies (2) for all test functions and this is the place where Hilbert spaces come into play.
    [Show full text]
  • Linear Spaces
    Chapter 2 Linear Spaces Contents FieldofScalars ........................................ ............. 2.2 VectorSpaces ........................................ .............. 2.3 Subspaces .......................................... .............. 2.5 Sumofsubsets........................................ .............. 2.5 Linearcombinations..................................... .............. 2.6 Linearindependence................................... ................ 2.7 BasisandDimension ..................................... ............. 2.7 Convexity ............................................ ............ 2.8 Normedlinearspaces ................................... ............... 2.9 The `p and Lp spaces ............................................. 2.10 Topologicalconcepts ................................... ............... 2.12 Opensets ............................................ ............ 2.13 Closedsets........................................... ............. 2.14 Boundedsets......................................... .............. 2.15 Convergence of sequences . ................... 2.16 Series .............................................. ............ 2.17 Cauchysequences .................................... ................ 2.18 Banachspaces....................................... ............... 2.19 Completesubsets ....................................... ............. 2.19 Transformations ...................................... ............... 2.21 Lineartransformations.................................. ................ 2.21
    [Show full text]
  • Chapter IX. Tensors and Multilinear Forms
    Notes c F.P. Greenleaf and S. Marques 2006-2016 LAII-s16-quadforms.tex version 4/25/2016 Chapter IX. Tensors and Multilinear Forms. IX.1. Basic Definitions and Examples. 1.1. Definition. A bilinear form is a map B : V V C that is linear in each entry when the other entry is held fixed, so that × → B(αx, y) = αB(x, y)= B(x, αy) B(x + x ,y) = B(x ,y)+ B(x ,y) for all α F, x V, y V 1 2 1 2 ∈ k ∈ k ∈ B(x, y1 + y2) = B(x, y1)+ B(x, y2) (This of course forces B(x, y)=0 if either input is zero.) We say B is symmetric if B(x, y)= B(y, x), for all x, y and antisymmetric if B(x, y)= B(y, x). Similarly a multilinear form (aka a k-linear form , or a tensor− of rank k) is a map B : V V F that is linear in each entry when the other entries are held fixed. ×···×(0,k) → We write V = V ∗ . V ∗ for the set of k-linear forms. The reason we use V ∗ here rather than V , and⊗ the⊗ rationale for the “tensor product” notation, will gradually become clear. The set V ∗ V ∗ of bilinear forms on V becomes a vector space over F if we define ⊗ 1. Zero element: B(x, y) = 0 for all x, y V ; ∈ 2. Scalar multiple: (αB)(x, y)= αB(x, y), for α F and x, y V ; ∈ ∈ 3. Addition: (B + B )(x, y)= B (x, y)+ B (x, y), for x, y V .
    [Show full text]