Homework 2 1. Let X and Y Be Hilbert Spaces Over C. Then a Sesquilinear Form H on X ×Y Is a Mapping H
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Baire Category Theorem and Uniform Boundedness Principle)
Functional Analysis (WS 19/20), Problem Set 3 (Baire Category Theorem and Uniform Boundedness Principle) Baire Category Theorem B1. Let (X; k·kX ) be an innite dimensional Banach space. Prove that X has uncountable Hamel basis. Note: This is Problem A2 from Problem Set 1. B2. Consider subset of bounded sequences 1 A = fx 2 l : only nitely many xk are nonzerog : Can one dene a norm on A so that it becomes a Banach space? Consider the same question with the set of polynomials dened on interval [0; 1]. B3. Prove that the set L2(0; 1) has empty interior as the subset of Banach space L1(0; 1). B4. Let f : [0; 1) ! [0; 1) be a continuous function such that for every x 2 [0; 1), f(kx) ! 0 as k ! 1. Prove that f(x) ! 0 as x ! 1. B5.( Uniform Boundedness Principle) Let (X; k · kX ) be a Banach space and (Y; k · kY ) be a normed space. Let fTαgα2A be a family of bounded linear operators between X and Y . Suppose that for any x 2 X, sup kTαxkY < 1: α2A Prove that . supα2A kTαk < 1 Uniform Boundedness Principle U1. Let F be a normed space C[0; 1] with L2(0; 1) norm. Check that the formula 1 Z n 'n(f) = n f(t) dt 0 denes a bounded linear functional on . Verify that for every , F f 2 F supn2N j'n(f)j < 1 but . Why Uniform Boundedness Principle is not satised in this case? supn2N k'nk = 1 U2.( pointwise convergence of operators) Let (X; k · kX ) be a Banach space and (Y; k · kY ) be a normed space. -
3 Bilinear Forms & Euclidean/Hermitian Spaces
3 Bilinear Forms & Euclidean/Hermitian Spaces Bilinear forms are a natural generalisation of linear forms and appear in many areas of mathematics. Just as linear algebra can be considered as the study of `degree one' mathematics, bilinear forms arise when we are considering `degree two' (or quadratic) mathematics. For example, an inner product is an example of a bilinear form and it is through inner products that we define the notion of length in n p 2 2 analytic geometry - recall that the length of a vector x 2 R is defined to be x1 + ... + xn and that this formula holds as a consequence of Pythagoras' Theorem. In addition, the `Hessian' matrix that is introduced in multivariable calculus can be considered as defining a bilinear form on tangent spaces and allows us to give well-defined notions of length and angle in tangent spaces to geometric objects. Through considering the properties of this bilinear form we are able to deduce geometric information - for example, the local nature of critical points of a geometric surface. In this final chapter we will give an introduction to arbitrary bilinear forms on K-vector spaces and then specialise to the case K 2 fR, Cg. By restricting our attention to thse number fields we can deduce some particularly nice classification theorems. We will also give an introduction to Euclidean spaces: these are R-vector spaces that are equipped with an inner product and for which we can `do Euclidean geometry', that is, all of the geometric Theorems of Euclid will hold true in any arbitrary Euclidean space. -
Representation of Bilinear Forms in Non-Archimedean Hilbert Space by Linear Operators
Comment.Math.Univ.Carolin. 47,4 (2006)695–705 695 Representation of bilinear forms in non-Archimedean Hilbert space by linear operators Toka Diagana This paper is dedicated to the memory of Tosio Kato. Abstract. The paper considers representing symmetric, non-degenerate, bilinear forms on some non-Archimedean Hilbert spaces by linear operators. Namely, upon making some assumptions it will be shown that if φ is a symmetric, non-degenerate bilinear form on a non-Archimedean Hilbert space, then φ is representable by a unique self-adjoint (possibly unbounded) operator A. Keywords: non-Archimedean Hilbert space, non-Archimedean bilinear form, unbounded operator, unbounded bilinear form, bounded bilinear form, self-adjoint operator Classification: 47S10, 46S10 1. Introduction Representing bounded or unbounded, symmetric, bilinear forms by linear op- erators is among the most attractive topics in representation theory due to its significance and its possible applications. Applications include those arising in quantum mechanics through the study of the form sum associated with the Hamil- tonians, mathematical physics, symplectic geometry, variational methods through the study of weak solutions to some partial differential equations, and many oth- ers, see, e.g., [3], [7], [10], [11]. This paper considers representing symmetric, non- degenerate, bilinear forms defined over the so-called non-Archimedean Hilbert spaces Eω by linear operators as it had been done for closed, positive, symmetric, bilinear forms in the classical setting, see, e.g., Kato [11, Chapter VI, Theo- rem 2.23, p. 331]. Namely, upon making some assumptions it will be shown that if φ : D(φ) × D(φ) ⊂ Eω × Eω 7→ K (K being the ground field) is a symmetric, non-degenerate, bilinear form, then there exists a unique self-adjoint (possibly unbounded) operator A such that (1.1) φ(u, v)= hAu, vi, ∀ u ∈ D(A), v ∈ D(φ) where D(A) and D(φ) denote the domains of A and φ, respectively. -
Majorana Spinors
MAJORANA SPINORS JOSE´ FIGUEROA-O'FARRILL Contents 1. Complex, real and quaternionic representations 2 2. Some basis-dependent formulae 5 3. Clifford algebras and their spinors 6 4. Complex Clifford algebras and the Majorana condition 10 5. Examples 13 One dimension 13 Two dimensions 13 Three dimensions 14 Four dimensions 14 Six dimensions 15 Ten dimensions 16 Eleven dimensions 16 Twelve dimensions 16 ...and back! 16 Summary 19 References 19 These notes arose as an attempt to conceptualise the `symplectic Majorana{Weyl condition' in 5+1 dimensions; but have turned into a general discussion of spinors. Spinors play a crucial role in supersymmetry. Part of their versatility is that they come in many guises: `Dirac', `Majorana', `Weyl', `Majorana{Weyl', `symplectic Majorana', `symplectic Majorana{Weyl', and their `pseudo' counterparts. The tra- ditional physics approach to this topic is a mixed bag of tricks using disparate aspects of representation theory of finite groups. In these notes we will attempt to provide a uniform treatment based on the classification of Clifford algebras, a work dating back to the early 60s and all but ignored by the theoretical physics com- munity. Recent developments in superstring theory have made us re-examine the conditions for the existence of different kinds of spinors in spacetimes of arbitrary signature, and we believe that a discussion of this more uniform approach is timely and could be useful to the student meeting this topic for the first time or to the practitioner who has difficulty remembering the answer to questions like \when do symplectic Majorana{Weyl spinors exist?" The notes are organised as follows. -
A Symplectic Banach Space with No Lagrangian Subspaces
transactions of the american mathematical society Volume 273, Number 1, September 1982 A SYMPLECTIC BANACHSPACE WITH NO LAGRANGIANSUBSPACES BY N. J. KALTON1 AND R. C. SWANSON Abstract. In this paper we construct a symplectic Banach space (X, Ü) which does not split as a direct sum of closed isotropic subspaces. Thus, the question of whether every symplectic Banach space is isomorphic to one of the canonical form Y X Y* is settled in the negative. The proof also shows that £(A") admits a nontrivial continuous homomorphism into £(//) where H is a Hilbert space. 1. Introduction. Given a Banach space E, a linear symplectic form on F is a continuous bilinear map ß: E X E -> R which is alternating and nondegenerate in the (strong) sense that the induced map ß: E — E* given by Û(e)(f) = ü(e, f) is an isomorphism of E onto E*. A Banach space with such a form is called a symplectic Banach space. It can be shown, by essentially the argument of Lemma 2 below, that any symplectic Banach space can be renormed so that ß is an isometry. Any symplectic Banach space is reflexive. Standard examples of symplectic Banach spaces all arise in the following way. Let F be a reflexive Banach space and set E — Y © Y*. Define the linear symplectic form fiyby Qy[(^. y% (z>z*)] = z*(y) ~y*(z)- We define two symplectic spaces (£,, ß,) and (E2, ß2) to be equivalent if there is an isomorphism A : Ex -» E2 such that Q2(Ax, Ay) = ß,(x, y). A. Weinstein [10] has asked the question whether every symplectic Banach space is equivalent to one of the form (Y © Y*, üy). -
Properties of Bilinear Forms on Hilbert Spaces Related to Stability Properties of Certain Partial Differential Operators
JOURNAL OF hlATHEhlATICAL ANALYSIS AND APPLICATIONS 20, 124-144 (1967) Properties of Bilinear Forms on Hilbert Spaces Related to Stability Properties of Certain Partial Differential Operators N. SAUER National Research Institute for Mathematical Sciences, Pretoria, South Africa Submitted by P. Lax INTRODUCTION The continuous dependence of solutions of positive definite, self-adjoint partial differential equations on the domain was investigated by Babugka [I], [2]. In a recent paper, [3], Babugka and Vjibornjr also studied the continuous dependence of the eigenvalues of strongly-elliptic, self-adjoint partial dif- ferential operators on the domain. It is the aim of this paper to study formalisms in abstract Hilbert space by means of which results on various types of continuous dependences can be obtained. In the case of the positive definite operator it is found that the condition of self-adjointness can be dropped. The properties of operators of this type are studied in part II. In part III the behavior of the eigenvalues of a self- adjoint elliptic operator under various “small changes” is studied. In part IV some applications to partial differential operators and variational theory are sketched. I. BASIC: CONCEPTS AND RESULTS Let H be a complex Hilbert space. We use the symbols X, y, z,... for vectors in H and the symbols a, b, c,... for scalars. The inner product in H is denoted by (~,y) and the norm by I( .v 11 = (x, x)llz. Weak and strong convergence of a sequence (x~} to an element x0 E H are denoted by x’, - x0 and x’, + x,, , respectively. A functional f on H is called: (i) continuous if it is continuous in the strong topology on H, (ii) weakly continuous (w.-continuous) if it is continuous in the weak topology on H. -
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 . -
Compactification of Classical Groups
COMMUNICATIONS IN ANALYSIS AND GEOMETRY Volume 10, Number 4, 709-740, 2002 Compactification of Classical Groups HONGYU HE 0. Introduction. 0.1. Historical Notes. Compactifications of symmetric spaces and their arithmetic quotients have been studied by many people from different perspectives. Its history went back to E. Cartan's original paper ([2]) on Hermitian symmetric spaces. Cartan proved that a Hermitian symmetric space of noncompact type can be realized as a bounded domain in a complex vector space. The com- pactification of Hermitian symmetric space was subsequently studied by Harish-Chandra and Siegel (see [16]). Thereafter various compactifications of noncompact symmetric spaces in general were studied by Satake (see [14]), Furstenberg (see [3]), Martin (see [4]) and others. These compact- ifications are more or less of the same nature as proved by C. Moore (see [11]) and by Guivarc'h-Ji-Taylor (see [4]). In the meanwhile, compacti- fication of the arithmetic quotients of symmetric spaces was explored by Satake (see [15]), Baily-Borel (see [1]), and others. It plays a significant role in the theory of automorphic forms. One of the main problems involved is the analytic properties of the boundary under the compactification. In all these compactifications that have been studied so far, the underlying compact space always has boundary. For a more detailed account of these compactifications, see [16] and [4]. In this paper, motivated by the author's work on the compactification of symplectic group (see [9]), we compactify all the classical groups. One nice feature of our compactification is that the underlying compact space is a symmetric space of compact type. -
Forms on Inner Product Spaces
FORMS ON INNER PRODUCT SPACES MARIA INFUSINO PROSEMINAR ON LINEAR ALGEBRA WS2016/2017 UNIVERSITY OF KONSTANZ Abstract. This note aims to give an introduction on forms on inner product spaces and their relation to linear operators. After briefly recalling some basic concepts from the theory of linear operators on inner product spaces, we focus on the space of forms on a real or complex finite-dimensional vector space V and show that it is isomorphic to the space of linear operators on V . We also describe the matrix representation of a form with respect to an ordered basis of the space on which it is defined, giving special attention to the case of forms on finite-dimensional complex inner product spaces and in particular to Hermitian forms. Contents Introduction 1 1. Preliminaries 1 2. Sesquilinear forms on inner product spaces 3 3. Matrix representation of a form 4 4. Hermitian forms 6 Notes 7 References 8 FORMS ON INNER PRODUCT SPACES 1 Introduction In this note we are going to introduce the concept of forms on an inner product space and describe some of their main properties (c.f. [2, Section 9.2]). The notion of inner product is a basic notion in linear algebra which allows to rigorously in- troduce on any vector space intuitive geometrical notions such as the length of an element and the orthogonality between two of them. We will just recall this notion in Section 1 together with some basic examples (for more details on this structure see e.g. [1, Chapter 3], [2, Chapter 8], [3]). -
Fact Sheet Functional Analysis
Fact Sheet Functional Analysis Literature: Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen. Teubner, 1986. Knabner, P., Angermann, L.: Numerik partieller Differentialgleichungen. Springer, 2000. Triebel, H.: H¨ohere Analysis. Harri Deutsch, 1980. Dobrowolski, M.: Angewandte Funktionalanalysis, Springer, 2010. 1. Banach- and Hilbert spaces Let V be a real vector space. Normed space: A norm is a mapping k · k : V ! [0; 1), such that: kuk = 0 , u = 0; (definiteness) kαuk = jαj · kuk; α 2 R; u 2 V; (positive scalability) ku + vk ≤ kuk + kvk; u; v 2 V: (triangle inequality) The pairing (V; k · k) is called a normed space. Seminorm: In contrast to a norm there may be elements u 6= 0 such that kuk = 0. It still holds kuk = 0 if u = 0. Comparison of two norms: Two norms k · k1, k · k2 are called equivalent if there is a constant C such that: −1 C kuk1 ≤ kuk2 ≤ Ckuk1; u 2 V: If only one of these inequalities can be fulfilled, e.g. kuk2 ≤ Ckuk1; u 2 V; the norm k · k1 is called stronger than the norm k · k2. k · k2 is called weaker than k · k1. Topology: In every normed space a canonical topology can be defined. A subset U ⊂ V is called open if for every u 2 U there exists a " > 0 such that B"(u) = fv 2 V : ku − vk < "g ⊂ U: Convergence: A sequence vn converges to v w.r.t. the norm k · k if lim kvn − vk = 0: n!1 1 A sequence vn ⊂ V is called Cauchy sequence, if supfkvn − vmk : n; m ≥ kg ! 0 for k ! 1. -
Orthogonal, Symplectic and Unitary Representations of Finite Groups
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 12, Pages 4687{4727 S 0002-9947(01)02807-0 Article electronically published on June 27, 2001 ORTHOGONAL, SYMPLECTIC AND UNITARY REPRESENTATIONS OF FINITE GROUPS CARL R. RIEHM Abstract. Let K be a field, G a finite group, and ρ : G ! GL(V ) a linear representation on the finite dimensional K-space V . The principal problems considered are: I. Determine (up to equivalence) the nonsingular symmetric, skew sym- metric and Hermitian forms h : V × V ! K which are G-invariant. II. If h is such a form, enumerate the equivalence classes of representations of G into the corresponding group (orthogonal, symplectic or unitary group). III. Determine conditions on G or K under which two orthogonal, sym- plectic or unitary representations of G are equivalent if and only if they are equivalent as linear representations and their underlying forms are \isotypi- cally" equivalent. This last condition means that the restrictions of the forms to each pair of corresponding isotypic (homogeneous) KG-module components of their spaces are equivalent. We assume throughout that the characteristic of K does not divide 2jGj. Solutions to I and II are given when K is a finite or local field, or when K is a global field and the representation is \split". The results for III are strongest when the degrees of the absolutely irreducible representations of G are odd { for example if G has odd order or is an Abelian group, or more generally has a normal Abelian subgroup of odd index { and, in the case that K is a local or global field, when the representations are split. -
A PRIMER on SESQUILINEAR FORMS This Is an Alternative
A PRIMER ON SESQUILINEAR FORMS BRIAN OSSERMAN This is an alternative presentation of most of the material from x8.1, 8.2, 8.3, 8.4, 8.5 and 8.8 of Artin's book. Any terminology (such as sesquilinear form or complementary subspace) which is discussed here but not in Artin is optional for the course, and will not appear on homework or exams. 1. Sesquilinear forms n The dot product is an important tool for calculations in R . For instance, we can use it to n measure distance and angles. However, it doesn't come from just the vector space structure on R { to define it implicitly involves a choice of basis. In Math 67, you may have studied inner products on real vector spaces and learned that this is the general context in which the dot product arises. Now, we will make a more general study of the extra structure implicit in dot products and more generally inner products. This is the subject of bilinear forms. However, not all forms of interest are bilinear. When working with complex vector spaces, one often works with Hermitian forms, which toss in an extra complex conjugation. In order to handle both of these cases at once, we'll work in the context of sesquilinear forms. For convenience, we'll assume throughout that our vector spaces are finite dimensional. We first set up the background on field automorphisms. Definition 1.1. Let F be a field. An automorphism of F is a bijection from F to itself which preserves the operations of addition and multiplication.