DOCSLIB.ORG

Operators on Indefinite Inner Product Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Operators on Indefinite Inner Product Spaces OPERATORS ON INDEFINITE INNER PRODUCT SPACES Michael A Dritschel and James Rovnyak Department of Mathematics University of Virginia Charlottesville Virginia U S A Mathematics Subject Classication Primary B A Secondary C E A A A Contents Preface Intro duction Preliminaries and Notation A Prerequisites B Inner pro duct spaces C Kren spaces and fundamental decomp ositions D Example E Bo oks and exp ositions Lecture Kren Spaces and Op erators Lecture Julia Op erators and Contractions Addendum Adjoints of contractions Lecture Extension and Completion Problems Lecture The Schur Algorithm Lecture Repro ducing Kernel Pontryagin Spaces and Colligations Lecture Invariant Subspaces A The commutant of a selfadjoint op erator B Dual pairs of invariant subspaces for denitizable op erators C Compact p erturbations of denitizable op erators Addendum Nondenitizable op erators Bibliography Index 1 2 M A Dritschel and J Rovnyak Preface Kren space op erator theory has long b een a principal area of research in eastern mathematics and it is now enjoying a surge of interest in western countries as well The present lectures survey some recent metho ds and results which center around factorization prop erties of op erators Topics include the geometry of Kren spaces Pontryagin spaces isometric op erators contractions dilation and extension problems repro ducing kernel spaces op erator colligations denitizable op erators and invariant subspaces Factorization ideas lead to notions of defect and Julia op erators which are to ols for reducing the study of general op erators to the isometric and unitary cases where Hilb ert space metho ds can often b e used Connections with interp olation theory and repro ducing kernel spaces are explored Factorization of selfadjoint op erators also has applications in sp ectral theory and invariant subspace problems It is not p ossible to represent the eld in six lectures and the selection of topics is based on a particular viewp oint Roughly the viewp oint is that Kren spaces provide a natural setting for op erator theory We assume Hilb ert spaces or Pontryagin spaces only when there is some reason for it Insights into the Hilb ert space case come out of this approach which separates the eects of p ositivity of vectors and p ositivity of op erators Connections b etween op erator and spatial p ositivity then arise in the form of index formulas Hilb ert space metho ds nd new applications in this approach and op erator theory is enriched by concepts which originate from the indeniteness of inner pro ducts The list of omitted topics is long For example we do not discuss mappings of op erator spheres and domains of holomorphy connections with univalent func tions commutant lifting interp olation theory or complementation theory for Kren spaces Our bibliography is likewise limited and we regret omissions On any topic the bibliography needs to b e supplemented from the usual sources Authoritative literature notes may b e found in the b o oks of Azizov and Iokhvidov Bognar and Iokhvidov Kren and Langer Two other lecture series in this work complement our account Systems asp ects are treated in the lectures of M A Kaasho ek on StateSpace Theory of Rational Matrix Functions and Applications Applications to dierential op erators and re lated topics such as op erator p olynomials are covered in the lectures of Aad Dijksma and Heinz Langer on Operator Theory and Ordinary Dierential Operators We thank Peter Lancaster and the Program Committee for inviting us to con tribute this series of lectures We also thank The Fields Institute for their supp ort ecient organization and hospitality Both authors are supp orted by the National Science Foundation Intro duction Preliminaries and Notation 3 Intro duction Preliminaries and Notation A Prerequisites A Hilb ert space is a strictly p ositive inner pro duct space H over the complex numb ers which is complete in its norm metric We also include separability in the axioms for a Hilb ert space Some familiarity with Hilb ert spaces and op erator theory is presumed Otherwise the material covered in a rst year graduate course of analysis is sucient background For example see Halmos and Rudin B Inner pro duct spaces As we use the term an inner pro duct space is a complex vector space H together with a complexvalued function h i h i on H H which satises H these axioms linearity haf bg hi a hf hi b hg hi for all f g h H and a b C hg f i for all f g H symmetry hf g i The function h i is called an inner pro duct By the antispace of an inner pro d uct space H h i we mean the inner pro duct space H h i Suppressing inner pro ducts we sometimes write H for the antispace of an inner pro duct space H Thus h i h i H H In any inner pro duct space H we use the notions of subspace orthogonality and direct sum in the same manner as for Hilb ert spaces For example vectors f and g in an inner pro duct space H are called orthogonal if hf g i Elementary prop erties which follow as in the Hilb ert space case are taken for granted An inner pro duct h i on H is said to b e nondegenerate if the only vector f in H such that hf g i for all vectors g H is f In applications this is usually a consequence of stronger assumptions For example Hilb ert spaces Pontryagin spaces and Kren spaces dened b elow are nondegenerate inner pro duct spaces An inner pro duct h i on H is said to b e p ositive if hf f i for all f H negative if hf f i for all f H denite if it is either p ositive or negative indenite if it is neither p ositive nor negative and neutral if it is b oth p ositive and negative and hence the inner pro duct is identically zero by the p olarization identity A p ositive negative inner pro duct is called strictly p ositive strictly negative if hf f i only for f These terms are applied to the space H as well as the inner pro duct They are also applied to subspaces M of H if the restriction of the inner pro duct has the corresp onding prop erty 4 M A Dritschel and J Rovnyak The term op erator always refers to a linear transformation A complex numb er is identied with the op erator of multiplication by the identity op erator on a space H is denoted or We write dom A and ran A for the domain and range H of an op erator A Two inner pro duct spaces H h i and K h i are called isomorphic if H K there is a onetoone op erator A from H onto K such that hAf Ag i hf g i for K H all f g H Such an op erator A is said to b e an isomorphism C Kren spaces and fundamental decomp ositions The denition of a Kren space is made in such a way that the class of Kren spaces includes all Hilb ert spaces antispaces of Hilb ert spaces and is closed under the formation of orthogonal direct sums It is the smallest such class of inner pro duct spaces Denition By a Kren space we mean an inner product space H which can be expressed as an orthogonal direct sum H H H + where H is a Hilbert space and H is the antispace of a Hilbert space Any such + representation is cal led a fundamental decomposition of the space Given a Kren space H and fundamental decomposition by jHj we mean the Hilbert space obtained from by replacing H by its antispace jH j H jHj H jH j + The Hilbert space norm kk kk is cal led a norm for the Kren space H jHj In this situation the op erator J dened on H by H J f f f H + whenever f f f with f H is called a fundamental symmetry or + signature op erator for the space The relationship b etween the Kren space H h i H and the Hilb ert space jHj h i jHj is that H and jHj coincide as vector spaces and hf g i hJ f g i and hf g i hJ f g i H H jHj H H jHj for any vectors f and g Hilb ert spaces and antispaces of Hilb ert spaces are Kren spaces Excluding these cases a Kren space has innitely many fundamental decomp ositions and therefore innitely many asso ciated Hilb ert spaces jHj and norms kk kk We jHj compare these structures for two fundamental decomp ositions Intro duction Preliminaries and Notation 5 Theorem Let H be a Kren space with two fundamental decompositions H H H and H H H + + Dene X LH H by + + X f f f H + + + + if f f f f H Dene Y LjH j jH j by + + Y f f f jH j if f f f f H Then X and Y are onetoone and onto + Pro of Let us show that X has a closed graph For any z H the mapping f hf z i is a continuous linear functional on H and on H If f g h + n n n + H where f H g H and h H for all n and if f f in H n + n n n + + and g g in H then f g H in the inner pro duct of H for all n n n n + + in the inner pro duct of H In other words f g h with and so f g H + It follows that the graph of X is closed and so X is continuous h f g H by the closed graph theorem Supp ose that f H and f f f with f H Since + + + + hf f i f f f f f f + + 0 0 0 + + + + H + H H H + + kX f k kf k Therefore X is onetoone and has closed range + + and is orthogonal For supp ose that h b elongs to H The range of X is all of H + + to the range of X Let f H and f f f with f H Then h is + + + + and so h is orthogonal to f Therefore X f and f orthogonal to b oth f + + + h H and hh hi hh hi and so h We have now shown that X is 0 H H + onetoone and onto The assertions for Y may b e deduced similarly or by applying what we just proved to the antispace of H Corollary If H is a Kren space the numbers ind H dim H do not depend on the choice of decomposition
Load more

Recommended publications
  • Surjective Isometries on Grassmann Spaces 11
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of the Academy's Library SURJECTIVE ISOMETRIES ON GRASSMANN SPACES FERNANDA BOTELHO, JAMES JAMISON, AND LAJOS MOLNAR´ Abstract. Let H be a complex Hilbert space, n a given positive integer and let Pn(H) be the set of all projections on H with rank n. Under the condition dim H ≥ 4n, we describe the surjective isometries of Pn(H) with respect to the gap metric (the metric induced by the operator norm). 1. Introduction The study of isometries of function spaces or operator algebras is an important research area in functional analysis with history dating back to the 1930's. The most classical results in the area are the Banach-Stone theorem describing the surjective linear isometries between Banach spaces of complex- valued continuous functions on compact Hausdorff spaces and its noncommutative extension for general C∗-algebras due to Kadison. This has the surprising and remarkable consequence that the surjective linear isometries between C∗-algebras are closely related to algebra isomorphisms. More precisely, every such surjective linear isometry is a Jordan *-isomorphism followed by multiplication by a fixed unitary element. We point out that in the aforementioned fundamental results the underlying structures are algebras and the isometries are assumed to be linear. However, descriptions of isometries of non-linear structures also play important roles in several areas. We only mention one example which is closely related with the subject matter of this paper. This is the famous Wigner's theorem that describes the structure of quantum mechanical symmetry transformations and plays a fundamental role in the probabilistic aspects of quantum theory.
    [Show full text]
  • Weak Compactness in the Space of Operator Valued Measures and Optimal Control Nasiruddin Ahmed
    Weak Compactness in the Space of Operator Valued Measures and Optimal Control Nasiruddin Ahmed To cite this version: Nasiruddin Ahmed. Weak Compactness in the Space of Operator Valued Measures and Optimal Control. 25th System Modeling and Optimization (CSMO), Sep 2011, Berlin, Germany. pp.49-58, 10.1007/978-3-642-36062-6_5. hal-01347522 HAL Id: hal-01347522 https://hal.inria.fr/hal-01347522 Submitted on 21 Jul 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License WEAK COMPACTNESS IN THE SPACE OF OPERATOR VALUED MEASURES AND OPTIMAL CONTROL N.U.Ahmed EECS, University of Ottawa, Ottawa, Canada Abstract. In this paper we present a brief review of some important results on weak compactness in the space of vector valued measures. We also review some recent results of the author on weak compactness of any set of operator valued measures. These results are then applied to optimal structural feedback control for deterministic systems on infinite dimensional spaces. Keywords: Space of Operator valued measures, Countably additive op- erator valued measures, Weak compactness, Semigroups of bounded lin- ear operators, Optimal Structural control.
    [Show full text]
  • 1 Lecture 09
    Notes for Functional Analysis Wang Zuoqin (typed by Xiyu Zhai) September 29, 2015 1 Lecture 09 1.1 Equicontinuity First let's recall the conception of \equicontinuity" for family of functions that we learned in classical analysis: A family of continuous functions defined on a region Ω, Λ = ffαg ⊂ C(Ω); is called an equicontinuous family if 8 > 0; 9δ > 0 such that for any x1; x2 2 Ω with jx1 − x2j < δ; and for any fα 2 Λ, we have jfα(x1) − fα(x2)j < . This conception of equicontinuity can be easily generalized to maps between topological vector spaces. For simplicity we only consider linear maps between topological vector space , in which case the continuity (and in fact the uniform continuity) at an arbitrary point is reduced to the continuity at 0. Definition 1.1. Let X; Y be topological vector spaces, and Λ be a family of continuous linear operators. We say Λ is an equicontinuous family if for any neighborhood V of 0 in Y , there is a neighborhood U of 0 in X, such that L(U) ⊂ V for all L 2 Λ. Remark 1.2. Equivalently, this means if x1; x2 2 X and x1 − x2 2 U, then L(x1) − L(x2) 2 V for all L 2 Λ. Remark 1.3. If Λ = fLg contains only one continuous linear operator, then it is always equicontinuous. 1 If Λ is an equicontinuous family, then each L 2 Λ is continuous and thus bounded. In fact, this boundedness is uniform: Proposition 1.4. Let X,Y be topological vector spaces, and Λ be an equicontinuous family of linear operators from X to Y .
    [Show full text]
  • Math 259A Lecture 7 Notes
    Math 259A Lecture 7 Notes Daniel Raban October 11, 2019 1 WO and SO Continuity of Linear Functionals and The Pre-Dual of B 1.1 Weak operator and strong operator continuity of linear functionals Lemma 1.1. Let X be a vector space with seminorms p1; : : : ; pn. Let ' : X ! C be a Pn linear functional such that j'(x)j ≤ i=1 pi(x) for all x 2 X. Then there exist linear P functionals '1;:::;'n : X ! C such that ' = i 'i with j'i(x)j ≤ pi(x) for all x 2 X and for all i. Proof. Let D = fx~ = (x; : : : ; x): x 2 Xg ⊆ Xn, which is a vector subspace. On Xn, n P take p((xi)i=1) = i pi(xi). We also have a linear map' ~ : D ! C given by' ~(~x) = '(x). This map satisfies j~(~x)j ≤ p(~x). By the Hahn-Banach theorem, there exists an n ∗ extension 2 (X ) of' ~ such that j (x1; : : : ; xn)j ≤ p(x1; : : : ; xn). Now define 'k(x) := (0; : : : ; x; 0;::: ), where the x is in the k-th position. Theorem 1.1. Let ' : B! C be linear. ' is weak operator continuous if and only if it is it is strong operator continuous. Proof. We only need to show that if ' is strong operator continuous, then it is weak Pn operator continuous. So assume there exist ξ1; : : : ; ξn 2 X such that j'(x)j ≤ i=1 kxξik P for all x 2 B. By the lemma, we can split ' = 'k, such that j'k(x)j ≤ kxξkk for all x and k.
    [Show full text]
  • Derivations on Metric Measure Spaces
    Derivations on Metric Measure Spaces by Jasun Gong A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2008 Doctoral Committee: Professor Mario Bonk, Chair Professor Alexander I. Barvinok Professor Juha Heinonen (Deceased) Associate Professor James P. Tappenden Assistant Professor Pekka J. Pankka “Or se’ tu quel Virgilio e quella fonte che spandi di parlar si largo fiume?” rispuos’io lui con vergognosa fronte. “O de li altri poeti onore e lume, vagliami ’l lungo studio e ’l grande amore che m’ha fatto cercar lo tuo volume. Tu se’ lo mio maestro e ’l mio autore, tu se’ solo colui da cu’ io tolsi lo bello stilo che m’ha fatto onore.” [“And are you then that Virgil, you the fountain that freely pours so rich a stream of speech?” I answered him with shame upon my brow. “O light and honor of all other poets, may my long study and the intense love that made me search your volume serve me now. You are my master and my author, you– the only one from whom my writing drew the noble style for which I have been honored.”] from the Divine Comedy by Dante Alighieri, as translated by Allen Mandelbaum [Man82]. In memory of Juha Heinonen, my advisor, teacher, and friend. ii ACKNOWLEDGEMENTS This work was inspired and influenced by many people. I first thank my parents, Ping Po Gong and Chau Sim Gong for all their love and support. They are my first teachers, and from them I learned the value of education and hard work.
    [Show full text]
  • The Banach-Alaoglu Theorem for Topological Vector Spaces
    The Banach-Alaoglu theorem for topological vector spaces Christiaan van den Brink a thesis submitted to the Department of Mathematics at Utrecht University in partial fulfillment of the requirements for the degree of Bachelor in Mathematics Supervisor: Fabian Ziltener date of submission 06-06-2019 Abstract In this thesis we generalize the Banach-Alaoglu theorem to topological vector spaces. the theorem then states that the polar, which lies in the dual space, of a neighbourhood around zero is weak* compact. We give motivation for the non-triviality of this theorem in this more general case. Later on, we show that the polar is sequentially compact if the space is separable. If our space is normed, then we show that the polar of the unit ball is the closed unit ball in the dual space. Finally, we introduce the notion of nets and we use these to prove the main theorem. i ii Acknowledgments A huge thanks goes out to my supervisor Fabian Ziltener for guiding me through the process of writing a bachelor thesis. I would also like to thank my girlfriend, family and my pet who have supported me all the way. iii iv Contents 1 Introduction 1 1.1 Motivation and main result . .1 1.2 Remarks and related works . .2 1.3 Organization of this thesis . .2 2 Introduction to Topological vector spaces 4 2.1 Topological vector spaces . .4 2.1.1 Definition of topological vector space . .4 2.1.2 The topology of a TVS . .6 2.2 Dual spaces . .9 2.2.1 Continuous functionals .
    [Show full text]
  • Course Structure for M.Sc. in Mathematics (Academic Year 2019 − 2020)
    Course Structure for M.Sc. in Mathematics (Academic Year 2019 − 2020) School of Physical Sciences, Jawaharlal Nehru University 1 Contents 1 Preamble 3 1.1 Minimum eligibility criteria for admission . .3 1.2 Selection procedure . .3 2 Programme structure 4 2.1 Overview . .4 2.2 Semester wise course distribution . .4 3 Courses: core and elective 5 4 Details of the core courses 6 4.1 Algebra I .........................................6 4.2 Real Analysis .......................................8 4.3 Complex Analysis ....................................9 4.4 Basic Topology ...................................... 10 4.5 Algebra II ......................................... 11 4.6 Measure Theory .................................... 12 4.7 Functional Analysis ................................... 13 4.8 Discrete Mathematics ................................. 14 4.9 Probability and Statistics ............................... 15 4.10 Computational Mathematics ............................. 16 4.11 Ordinary Differential Equations ........................... 18 4.12 Partial Differential Equations ............................. 19 4.13 Project ........................................... 20 5 Details of the elective courses 21 5.1 Number Theory ..................................... 21 5.2 Differential Topology .................................. 23 5.3 Harmonic Analysis ................................... 24 5.4 Analytic Number Theory ............................... 25 5.5 Proofs ........................................... 26 5.6 Advanced Algebra ...................................
    [Show full text]
  • Spectral Radii of Bounded Operators on Topological Vector Spaces
    SPECTRAL RADII OF BOUNDED OPERATORS ON TOPOLOGICAL VECTOR SPACES VLADIMIR G. TROITSKY Abstract. In this paper we develop a version of spectral theory for bounded linear operators on topological vector spaces. We show that the Gelfand formula for spectral radius and Neumann series can still be naturally interpreted for operators on topological vector spaces. Of course, the resulting theory has many similarities to the conventional spectral theory of bounded operators on Banach spaces, though there are several im- portant differences. The main difference is that an operator on a topological vector space has several spectra and several spectral radii, which fit a well-organized pattern. 0. Introduction The spectral radius of a bounded linear operator T on a Banach space is defined by the Gelfand formula r(T ) = lim n T n . It is well known that r(T ) equals the n k k actual radius of the spectrum σ(T ) p= sup λ : λ σ(T ) . Further, it is known −1 {| | ∈ } ∞ T i that the resolvent R =(λI T ) is given by the Neumann series +1 whenever λ − i=0 λi λ > r(T ). It is natural to ask if similar results are valid in a moreP general setting, | | e.g., for a bounded linear operator on an arbitrary topological vector space. The author arrived to these questions when generalizing some results on Invariant Subspace Problem from Banach lattices to ordered topological vector spaces. One major difficulty is that it is not clear which class of operators should be considered, because there are several non-equivalent ways of defining bounded operators on topological vector spaces.
    [Show full text]
  • Weak Operator Topology, Operator Ranges and Operator Equations Via Kolmogorov Widths
    Weak operator topology, operator ranges and operator equations via Kolmogorov widths M. I. Ostrovskii and V. S. Shulman Abstract. Let K be an absolutely convex infinite-dimensional compact in a Banach space X . The set of all bounded linear operators T on X satisfying TK ⊃ K is denoted by G(K). Our starting point is the study of the closure WG(K) of G(K) in the weak operator topology. We prove that WG(K) contains the algebra of all operators leaving lin(K) invariant. More precise results are obtained in terms of the Kolmogorov n-widths of the compact K. The obtained results are used in the study of operator ranges and operator equations. Mathematics Subject Classification (2000). Primary 47A05; Secondary 41A46, 47A30, 47A62. Keywords. Banach space, bounded linear operator, Hilbert space, Kolmogorov width, operator equation, operator range, strong operator topology, weak op- erator topology. 1. Introduction Let K be a subset in a Banach space X . We say (with some abuse of the language) that an operator D 2 L(X ) covers K, if DK ⊃ K. The set of all operators covering K will be denoted by G(K). It is a semigroup with a unit since the identity operator is in G(K). It is easy to check that if K is compact then G(K) is closed in the norm topology and, moreover, sequentially closed in the weak operator topology (WOT). It is somewhat surprising that for each absolutely convex infinite- dimensional compact K the WOT-closure of G(K) is much larger than G(K) itself, and in many cases it coincides with the algebra L(X ) of all operators on X .
    [Show full text]
  • Bounded Operator - Wikipedia, the Free Encyclopedia
    Bounded operator - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Bounded_operator Bounded operator From Wikipedia, the free encyclopedia In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in X. In other words, there exists some M > 0 such that for all v in X The smallest such M is called the operator norm of L. A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless Y is the zero vector space. Rather, a bounded linear operator is a locally bounded function. A linear operator on a metrizable vector space is bounded if and only if it is continuous. Contents 1 Examples 2 Equivalence of boundedness and continuity 3 Linearity and boundedness 4 Further properties 5 Properties of the space of bounded linear operators 6 Topological vector spaces 7 See also 8 References Examples Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrix. Many integral transforms are bounded linear operators. For instance, if is a continuous function, then the operator defined on the space of continuous functions on endowed with the uniform norm and with values in the space with given by the formula is bounded.
    [Show full text]
  • Chapter 6 Linear Transformations and Operators
    Chapter 6 Linear Transformations and Operators 6.1 The Algebra of Linear Transformations Theorem 6.1.1. Let V and W be vector spaces over the field F . Let T and U be two linear transformations from V into W . The function (T + U) defined pointwise by (T + U)(v) = T v + Uv is a linear transformation from V into W . Furthermore, if s F , the function (sT ) ∈ defined by (sT )(v) = s (T v) is also a linear transformation from V into W . The set of all linear transformation from V into W , together with the addition and scalar multiplication defined above, is a vector space over the field F . Proof. Suppose that T and U are linear transformation from V into W . For (T +U) defined above, we have (T + U)(sv + w) = T (sv + w) + U (sv + w) = s (T v) + T w + s (Uv) + Uw = s (T v + Uv) + (T w + Uw) = s(T + U)v + (T + U)w, 127 128 CHAPTER 6. LINEAR TRANSFORMATIONS AND OPERATORS which shows that (T + U) is a linear transformation. Similarly, we have (rT )(sv + w) = r (T (sv + w)) = r (s (T v) + (T w)) = rs (T v) + r (T w) = s (r (T v)) + rT (w) = s ((rT ) v) + (rT ) w which shows that (rT ) is a linear transformation. To verify that the set of linear transformations from V into W together with the operations defined above is a vector space, one must directly check the conditions of Definition 3.3.1. These are straightforward to verify, and we leave this exercise to the reader.
    [Show full text]
  • Linear Algebra Review
    1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic facts about real and complex numbers. In particular, both R and C are fields; they satisfy the field axioms. For p z = x + iy 2 C, the modulus, i.e. jzj = x2 + y2 ≥ 0, represents the distance from z to the origin in the complex plane. (As such, it coincides with the absolute value for real z.) For z = x + iy 2 C, complex conjugation, i.e. z = x − iy, represents reflection about the x-axis in the complex plane. It will also be important that both R and C are complete, as metric spaces, when equipped with the metric d(z; w) = jz − wj. 1.1. Vector Spaces. One of the most important notions in this course is that of a vector space. Although vector spaces can be defined over any field, we will (by and large) restrict our attention to fields F 2 fR; Cg. The following definition is fundamental. Definition 1.1. Let F be a field. A vector space V over F is a non-empty set V (the elements of V are called vectors) over a field F (the elements of F are called scalars) equipped with two operations: i) To each pair u; v 2 V , there exists a unique element u + v 2 V . This operation is called vector addition. ii) To each u 2 V and α 2 F, there exists a unique element αu 2 V .
    [Show full text]