DOCSLIB.ORG

Appendix a Linear Operators in Banach Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Appendix a Linear Operators in Banach Spaces (5) Appendix A Linear operators in Banach Spaces The goal of this appendix is to recall fundamental results on linear operators in Banach and Hilbert spaces; see, e.g., Aubin [15], Brezis [56], Lax [166], Rudin [206], Yosida [251], Zeidler [252] for further reading. The results collected in this appendix provide a theoretical framework for the mathematical analysis of the finite element method. A.1 Banach and Hilbert spaces To stay general, we consider complex vector spaces, i.e., vector spaces over the field C of complex numbers. The case of real vector spaces is recovered by replacing the field C by R, by removing the real part symbol ( ) and the complex conjugate symbol , and by interpreting the symbol asℜ the· absolute value instead of the modulus.· Some additional minor adaptations|·| (regarding duality and inner products) are indicated whenever relevant. A.1.1 Banach spaces Let V be a complex vector space. Definition A.1 (Norm). A norm on V is a map V : V [0, ) satisfy- ing the following three properties: k·k → ∞ (i) Definiteness: v =0 v =0. k kV ⇐⇒ (ii) 1-homogeneity: λv V = λ v V , for all λ C and all v V . (iii) Triangle inequality:k k v + |w|k k v + w ∈ , for all v, w∈ V . k kV ≤k kV k kV ∈ A seminorm on V is a map : V [0, ) satisfying only (ii) and (iii). |·|V → ∞ Definition A.2 (Banach space). Equip V with a norm V ; then V is called a Banach space if every Cauchy sequence (with respectk·k to the metric d(x, y)= x y ) in V has a limit in V . k − kV 424 Appendix A. Linear operators in Banach Spaces Definition A.3 (Equivalent norms). Two norms V,1 and V,2 are said to be equivalent on V if there exists a positive real numberk·k c suchk·k that 1 c v v c− v , v V. (A.1) k kV,2 ≤k kV,1 ≤ k kV,2 ∀ ∈ One readily verifies that when (A.1) holds, V is a Banach space for the norm iff it is a Banach space for the norm . k·kV,1 k·kV,2 Remark A.4 (Finite dimension). If the vector space V is finite-dimensional, all the norms in V are equivalent. This result is false in infinite-dimensional vector spaces. Actually, the unit ball in V is a compact set (for the norm topology) if and only if V is finite-dimensional; see Brezis [56, Thm. 6.5], Lax [166, 5.2]. § ⊓⊔ A.1.2 Bounded linear maps Let V, W be complex vector spaces. A map A : V W is said to be linear → if A(v1 + v2) = A(v1)+ A(v2) for all v1, v2 V and A(λv) = λA(v) for all λ C and all v V . To alleviate the notation,∈ we simply write Av instead ∈ ∈ of A(v). Whenever V and W can be equipped with norms V and W , respectively, the linear map A is said to be bounded or continuousk·k if k·k Av W A (V ;W ) := sup k k < . (A.2) k kL v V v V ∞ ∈ k k In this book, we systematically abuse the notation by implicitly assuming that the argument in this type of supremum or infimum is nonzero. Proposition A.5 ( (V ; W )). The complex vector space composed of the L bounded linear maps from V to W is denoted (V ; W ). The map (V ;W ) defined by (A.2) is indeed a norm on (V ; W ),L and if W is a Banachk·kL space, then (V ; W ) equipped with this normL is also a Banach space. L Proof. See Rudin [206, p. 87], Yosida [251, p. 111]. ⊓⊔ Example A.6 (Continuous embedding). Assume that V W and that ⊂ there is a real number c such that v W c v V for all v V . Then the embedding of V into W is continuous.k Wek say≤ thatk kV is continuously∈ embedded . into W and we write V ֒ W → ⊓⊔ Whenever V and W are Banach spaces, maps in (V ; W ) are often called operators. The following result (often called UniformL Boundedness Principle) is a useful tool to study the limit of a sequence of operators. Theorem A.7 (Banach–Steinhaus). Let V, W be Banach spaces and let Ai i I be a family (not necessarily countable) in (V ; W ). Assume that { } ∈ L supi I Aiv W is a finite number for all v V . Then, there is a real number C such∈ k thatk ∈ sup Aiv W C v V , v V. (A.3) i I k k ≤ k k ∀ ∈ ∈ Appendices 425 Proof. See Brezis [56, p. 32], Lax [166, Chap. 10]. ⊓⊔ Corollary A.8 (Pointwise convergence). Let V, W be Banach spaces. Let (An)n N be a sequence in (V ; W ) such that, for all v V , the sequence ∈ L ∈ (Anv)n N converges as n to a limit in W denoted Av (this means that ∈ → ∞ the sequence (An)n N converges pointwise to A). Then, the following holds: ∈ (i) supn N An (V ;W ) < ; (ii) A ∈ (Vk; Wk)L; ∞ ∈ L (iii) A (V ;W ) lim infn An (V ;W ). k kL ≤ →∞ k kL Proof. Statement (i) is just a consequence of Theorem A.7. Owing to (A.3), we infer that A v C v for all v V and all n N. Letting n , k n kW ≤ k kV ∈ ∈ → ∞ we obtain that Av W C v V , and since A is obviously linear, we infer that statement (ii)k holds.k ≤ Finally,k k statement (iii) results from the fact that Anv W An (V ;W ) v V for all v V and all n N. k k ≤k kL k k ∈ ∈ ⊓⊔ Remark A.9 (Uniform convergence on compact sets). Corollary A.8 does not claim that (An)n N converges to A in (V ; W ), i.e., uniformly on ∈ L bounded sets. However, a standard argument shows that (An)n N converges uniformly to A on compact sets. Indeed, let K V be a compact∈ set. Let ⊂ ǫ> 0. Set C := supn N An (V ;W ); C is finite owing to Corollary A.8(i). The ∈ k kL set K being compact, we infer that there is a finite set of points xi i I in K { 1} ∈ such that, for all v K, there is i I such that v x (3C)− ǫ. Owing ∈ ∈ k − ikV ≤ to the pointwise convergence of (An)n N to A, there is Ni such that, for all 1 ∈ n Ni, Anxi Axi W 3 ǫ. Using the triangle inequality and statement (iii)≥ above,k we infer− thatk ≤ A v Av A (v x ) + A x Ax + A(v x ) ǫ, k n − kW ≤k n − i kW k n i − ikW k − i kW ≤ for all v K and all n maxi I Ni. ∈ ≥ ∈ ⊓⊔ A.1.3 Compact operators Definition A.10 (Compact operator). Let V, W be Banach spaces. The operator T (V ; W ) is said to be compact if from every bounded sequence ∈ L (vn)n N in V one can extract a subsequence (vn )k N such that the sequence ∈ k ∈ (T vnk )k N converges in W ; equivalently, T maps the unit ball in V into a relatively∈ compact set in W . Theorem A.11 (Schauder). A bounded linear operator between Banach spaces is compact if and only if its adjoint is compact. Example A.12 (Compact embedding). Assume that V W and that the embedding of V into W is compact. Then from every bounded⊂ sequence (vn)n N in V one can extract a subsequence that converges in W . ∈ ⊓⊔ Proposition A.13 (Composition). Let W , X, Y , Z be four Banach spaces and let A (Z; Y ), K (Y ; X), B (X; W ). Assume that K is compact; then the operator∈ L B K∈ LA is compact.∈ L ◦ ◦ 426 Appendix A. Linear operators in Banach Spaces A.1.4 Duality Let V, W be complex vector spaces. The map A : V W is said to be → antilinear if A(v1 + v2)= Av1 + Av2 for all v1, v2 V and A(λv) = λAv for all λ C and all v V . The reason for considering∈ antilinear maps is that the weak∈ form of complex-valued∈ PDEs is derived by multiplying the PDE by the complex conjugate of a test function; therefore, the right-hand side of the weak formulation is an antilinear form. Alternatively, one could work with linear maps only upon noticing that if A is an antilinear map, then A (defined by Av := Av for all v V ) is a linear map. ∈ Definition A.14 (Dual space). The dual space of V is denoted V ′ and is composed of the bounded antilinear forms from V to C. An element A V ′ is called bounded antilinear form, and its action on an element v V is denoted∈ ∈ either Av or A, v ′ . h iV ,V Owing to Proposition A.5, V ′ is a Banach space with the norm Av A, v V ′,V A V ′ = sup | | = sup |h i |, A V ′. (A.4) k k v V v V v V v V ∀ ∈ ∈ k k ∈ k k In the real case, the absolute value can be omitted from the numerators since v can be considered. In the complex case, the modulus can be replaced by the± real part since v can be multiplied by any unit complex number. Theorem A.15 (Hahn–Banach). Let V be a normed vector space over C and let W be a subspace of V .
Load more

Recommended publications
  • Math212a1403 the Spectral Theorem for Compact Self-Adjoint Operators
    Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. Math212a1403 The spectral theorem for compact self-adjoint operators. Shlomo Sternberg Sept. 9, 2014 Shlomo Sternberg Math212a1403 The spectral theorem for compact self-adjoint operators. Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. 1 Review of last lecture. 2 The Riesz representation theorem. 3 Bessel's inequality. 4 Self-adjoint transformations. Non-negative self-adjoint transformations. 5 Compact self-adjoint transformations. 6 The spectral theorem for compact self-adjoint operators. 7 Fourier's Fourier series. Proof by integration by parts. Shlomo Sternberg Math212a1403 The spectral theorem for compact self-adjoint operators. This w is characterized as being the unique element of M which minimizes kv − xk; x 2 M. The idea of the proof was to use the parallelogram law to conclude that if fxng is a sequence of elements of M such that kv − xnk approaches the greatest lower bound of kv − xk; x 2 M then fxng converges to the desired w, The map v 7! w is called orthogonal projection of H onto M and is denoted by πM . Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. Orthogonal projection onto a complete subspace.
    [Show full text]
  • Hilbert Spaces
    C H A P T E R 12 Hilbert Spaces The material to be presented in this chapter is essential for all advanced work in physics and analysis. We have attempted to present several relatively diffi- cult theorems in sufficient detail that they are readily understandable by readers with less background than normally might be required for such results. However, we assume that the reader is quite familiar with the contents of Appendices A and B, and we will frequently refer to results from these appendices. Essentially, this chapter serves as an introduction to the theory of infinite-dimensional vector spaces. Throughout this chapter we let E, F and G denote normed vector spaces over the real or complex number fields only. 12.1 MATHEMATICAL PRELIMINARIES This rather long first section presents the elementary properties of limits and continuous functions. While most of this material properly falls under the heading of analysis, we do not assume that the reader has already had such a course. However, if these topics are familiar, then the reader should briefly scan the theorems of this section now, and return only for details if and when it becomes necessary. For ease of reference, we briefly repeat some of our earlier definitions and results. By a norm on a vector space E, we mean a mapping ˜ ˜: E ‘ ® satis- fying: 619 620 HILBERT SPACES (N1) ˜u˜ ˘ 0 for every u ∞ E and ˜u˜ = 0 if and only if u = 0 (positive definiteness). (N2) ˜cu˜ = \c\ ˜u˜ for every u ∞ E and c ∞ F. (N3) ˜u + v˜ ¯ ˜u˜ + ˜v˜ (triangle inequality).
    [Show full text]
  • On the Kazhdan-Lusztig Polynomials for Affine Weyl Groups
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ADVANCES IN MATHEMATICS 55, 103-130 (1985) On the Kazhdan-Lusztig Polynomials for Affine Weyl Groups SHIN-ICHI KATO * Department of Mathematics, Faculty of Science. University of Tokyo, Hongo, Tokyo, 113 Japan In 191, Kazhdan and Lusztig associated to a Coxeter group W certain polynomials P,,, (y, w E W). In the case where W is a Weyl group, these polynomials (or their values at 1, P,,,,(l)) give multiplicities of irreducible constituents in the Jordan-Holder series of Verma modules for a complex semisimple Lie algebra corresponding to W (see [ 3,4]). Also, in the case where W is an affine Weyl group, it has been conjectured by Lusztig [lo] that these polynomials or their values P,,,(l) enter the character formula of irreducible rational representations for a semisimple algebraic group (corresponding to W) over an algebraically closed field of prime charac- teristic. The purpose of this paper is to give some formula (4.2, see also 4.10) concerning the Kazhdan-Lusztig polynomials P,,, for affine Weyl groups which appear in this Lusztig conjecture; namely P,,, for y, w “dominant.” (This is a generalization of the q-analogue of Kostant’s weight multiplicity formula in [8].) As a corollary of the formula, we shall show that the Lusztig conjecture above is consistent with the Steinberg tensor product theorem [ 15 1. To be more precise, let P be the weight lattice of a root system R (II) R ‘, a positive root system).
    [Show full text]
  • Introduction to Coherent Quantization
    Introduction to coherent quantization Arnold Neumaier Fakult¨at f¨ur Mathematik, Universit¨at Wien Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria email: [email protected] WWW: http://www.mat.univie.ac.at/~neum Arash Ghaani Farashahi Department of Pure Mathematics, School of Mathematics Faculty of Mathematics and Physical Sciences, University of Leeds Leeds, LS2 9JT, United Kingdom email: [email protected] email: [email protected] WWW: https://sites.google.com/site/ghaanifarashahi/ August 16, 2021 arXiv:1804.01400v2 Abstract. This paper is one of a series of papers on coherent spaces and their appli- cations, defined in the recent book ’Coherent Quantum Mechanics’ by the first author. The paper studies coherent quantization – the way operators in the quantum space of a coherent space can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization, including the non-unitary case. Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties. The paper characterizes linear operators acting on the quantum space of a coherent space in terms of their coherent matrix elements. Coherent maps and associated symmetry groups for coherent spaces are introduced, and formulas are derived for the quantization arXiv:1804.01400v3 [math-ph] 16 Aug 2021 of coherent maps.
    [Show full text]
  • Complex Symmetric Operators and Applications
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 358, Number 3, Pages 1285–1315 S 0002-9947(05)03742-6 Article electronically published on May 26, 2005 COMPLEX SYMMETRIC OPERATORS AND APPLICATIONS STEPHAN RAMON GARCIA AND MIHAI PUTINAR Abstract. We study a few classes of Hilbert space operators whose matrix representations are complex symmetric with respect to a preferred orthonormal basis. The existence of this additional symmetry has notable implications and, in particular, it explains from a unifying point of view some classical results. We explore applications of this symmetry to Jordan canonical models, self- adjoint extensions of symmetric operators, rank-one unitary perturbations of the compressed shift, Darlington synthesis and matrix-valued inner functions, and free bounded analytic interpolation in the disk. 1. Introduction The simultaneous diagonalization and spectral analysis of two Hermitian forms goes back to the origins of Hilbert space theory and, in particular, to the spectral theorem for self-adjoint operators. Even today the language of forms is often used when dealing with unbounded operators (see [31, 43]). The similar theory for a Hermitian and a nondefinite sesquilinear form was motivated by the Hamiltonian mechanics of strings or continuous media models; from a mathematical point of view this theory leads to Hilbert spaces with a complex linear J-involution and the associated theory of J-unitary and J-contractive operators (see for instance [19, 31, 41]). Less studied, but not less important, is the simultaneous analysis of a pair consisting of a Hermitian form and a bilinear form; this framework has appeared quite early in function theory ([9, 48, 51]), functional analysis ([37]), and elasticity theory ([17]).
    [Show full text]
  • 5. SPINORS 5.1. Prologue. 5.2. Clifford Algebras and Their
    5. SPINORS 5.1. Prologue. 5.2. Clifford algebras and their representations. 5.3. Spin groups and spin representations. 5.4. Reality of spin modules. 5.5. Pairings and morphisms. 5.6. Image of the real spin group in the complex spin module. 5.7. Appendix: Some properties of the orthogonal groups. 5.1. Prologue. E. Cartan classified simple Lie algebras over C in his thesis in 1894, a classification that is nowadays done through the (Dynkin) diagrams. In 1913 he classified the irreducible finite dimensional representations of these algebras1. For any simple Lie algebra g Cartan's construction yields an irreducible representation canonically associated to each node of its diagram. These are the so-called fun- damental representations in terms of which all irreducible representations of g can be constructed using and subrepresentations. Indeed, if πj(1 j `) are the ⊗ ≤ ≤ fundamental representations and mj are integers 0, and if vj is the highest vector ≥ of πj, then the subrepresentation of m1 m` π = π⊗ ::: π⊗ 1 ⊗ ⊗ ` generated by m1 m` v = v⊗ ::: v⊗ 1 ⊗ ⊗ ` is irreducible with highest vector v, and every irreducible module is obtained in this manner uniquely. As is well-known, Harish-Chandra and Chevalley (independently) developed around 1950 a general method for obtaining the irreducible representa- tions without relying on case by case considerations as Cartan did. `+1 j If g = sl(` + 1) and V = C , then the fundamental module πj is Λ (V ), and all irreducible modules can be obtained by decomposing the tensor algebra over the 1 defining representation V . Similarly, for the symplectic Lie algebras, the decomposi- tion of the tensors over the defining representation gives all the irreducible modules.
    [Show full text]
  • Some New Directions in Teichmüller Theory
    Some New Directions in Teichmu¨ller Theory by Mark Greenfield A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2021 Doctoral Committee: Professor Lizhen Ji, Chair Professor Christine Aidala Professor Richard Canary Associate Professor Sarah Koch Professor Ralf Spatzier Mark Greenfield [email protected] ORCID iD: 0000-0001-7676-0557 c Mark Greenfield 2021 © All Rights Reserved ACKNOWLEDGEMENTS First and foremost I am grateful to my advisor, Lizhen Ji, for supporting me throughout my graduate career. He has introduced me to many different areas of mathematics, and always had confidence in my ability to make meaningful contri- butions. I would like to thank my doctoral committee for working with me through this process during a pandemic. I am indebted to many professors at the Univer- sity of Michigan, especially Sarah Koch and Richard Canary, for numerous helpful conversations. Dmitri Gekhtman deserves special mention, as Chapter V would not exist without our fruitful collaboration and friendship. I would like to acknowledge Bradley Zykoski for many fascinating mathematical and non-mathematical discus- sions over the years, and for providing helpful comments on early drafts of this thesis. There are numerous others who deserve to be listed here, but I hope a gen- eral expression of gratitude to the wonderful people in the department and beyond, including but not limited to my fellow graduate students, postdoctoral scholars, ad- ministrative staff, lecturers, friends from my undergraduate years, as well as my students, will suffice. Most of this work was completed while I was funded by the National Science Foundation Graduate Research Fellowship Program under Grant No.
    [Show full text]
  • Rigged Hilbert Space Theory for Hermitian and Quasi-Hermitian Observables
    Rigged Hilbert Space Theory for Hermitian and Quasi-Hermitian Observables Bachelor Thesis: Mathematics and Physics July 2019 Student: Lars van der Laan Faculty: University of Groningen FSE Primary Advisor Mathematics: Prof. A. Sterk Primary Advisor Physics: Prof. D. Boer Abstract In this thesis, we study whether the rigged Hilbert space formulation of Quantum Mechanics can be extended to quasi-hermitian operators. The necessary mathematics to understand rigged Hilbert spaces and their representations is presented. Using the mathematics, we give rigorous meaning to Dirac’s kets and bras, using the momentum operator as an example. The mathe- matical theory culminates with the nuclear spectral theorem(Gelfand-Maurin) which guarantees an eigenket decomposition for the wave functions. Next, we illustrate the theory by applying it to the quantum system defined by a potential barrier. We then introduce non-hermitian oper- ators of interest in Quantum Mechanics, specifically PT-symmetric Hamiltonians. It is known that PT-symmetric Hamiltonians are necessarily pseudo-hermitian. We restrict our attention to a subclass of pseudo-hermitian operators, the quasi-hermitian operators, and present an extension of the nuclear spectral theorem. The extension, under certain conditions on the quasi-hermitian operator and the metric operator, guarantees an eigenket decomposition for a dense subset of the Hilbert space on which the operator is defined. We discuss how the extension may help put quasi-hermitian quantum mechanics on a rigorous footing and discuss its drawbacks and possible extensions. Lastly, we present conditions under which a spectral decomposition of quasi-hermitian operators with respect to an operator-valued measure exists. This is an extension of the spectral decomposition of self-adjoint operators with respect to a projection-valued measure.
    [Show full text]
  • C –Algebras (A, K·Ka, ∗) and (B, K·Kb, ∗), We Want to Construct C∗–Norms on the Algebraic Tensor Product a ⊗ B
    C∗–Algebras Christian B¨ar1 and Christian Becker2 1 Institut f¨urMathematik, Universit¨atPotsdam, [email protected] 2 Institut f¨urMathematik, Universit¨atPotsdam, [email protected] In this chapter we will collect those basic concepts and facts related to C∗– algebras that will be needed later on. We give complete proofs. In Sections 1, 2, 3, and 6 we follow closely the presentation in [1]. For more information on C∗–algebras see e. g. [2, 4, 5, 7, 8]. 1 Basic definitions Definition 1. Let A be an associative C-algebra, let k · k be a norm on the C-vector space A, and let ∗ : A → A, a 7→ a∗, be a C-antilinear map. Then (A, k · k, ∗) is called a C∗–algebra, if (A, k · k) is complete and we have for all a, b ∈ A: 1. a∗∗ = a (∗ is an involution) 2. (ab)∗ = b∗a∗ 3. kabk ≤ kak kbk (submultiplicativity) 4. ka∗k = kak (∗ is an isometry) 5. ka∗ak = kak2 (C∗–property). A (not necessarily complete) norm on A satisfying conditions (1) to (5) is called a C∗–norm. Remark 1. Note that the Axioms 1–5 are not independent. For instance, Ax- iom 4 can easily be deduced from Axioms 1,3 and 5. Example 1. Let (H, (·, ·)) be a complex Hilbert space, let A = L(H) be the algebra of bounded operators on H. Let k · k be the operator norm, i. e., kak := sup kaxk. x∈H kxk=1 Let a∗ be the operator adjoint to a, i. e., 2 Christian B¨arand Christian Becker (ax, y) = (x, a∗y) for all x, y ∈ H.
    [Show full text]
  • Appendix a Banach and Hilbert Spaces A.1 Banach Spaces
    Appendix A Banach and Hilbert spaces The goal of this appendix is to recall basic results on Banach and Hilbert (JLG) Sept 15 2019 spaces. To stay general, we consider complex vector spaces, i.e., vector spaces over the field C of complex numbers. The case of real vector spaces is recov- ered by replacing the field C by R, by removing the real part symbol ( ) and the complex conjugate symbol , and by interpreting the symbol ℜ as· the absolute value instead of the modulus.· |·| A.1 Banach spaces Let V be a complex vector space. Definition A.1 (Norm). A norm on V is a map V : V R+ := [0, ) satisfying the following three properties: k·k → ∞ (i) Definiteness: [ v = 0 ] [ v = 0 ]. k kV ⇐⇒ (ii) 1-homogeneity: λv V = λ v V for all λ C and all v V . (iii) Triangle inequality:k k v + |w|k k v + w∈ for all v, w∈ V . k kV ≤k kV k kV ∈ For every norm V : V R+ := [0, ), the function d(x, y) := x y V , for all x, y V ,k·k is a metric→ (or distance).∞ k − k ∈ Remark A.2 (Definiteness). Item (i) can be slightly relaxed by requiring only that [ v V = 0] = [ v = 0 ], since the 1-homogeneity implies that [ v =0]= k[ kv = 0 ]. ⇒ ⇒ k kV ⊓⊔ Definition A.3 (Seminorm). A seminorm on V is a map V : V R+ satisfying only the statements (ii) and (iii) above, i.e., 1-homogeneity|·| and→ the triangle inequality Definition A.4 (Banach space).
    [Show full text]
  • 1 Yang-Mills Theory and a Superquadric
    1 Yang-Mills theory and a superquadric. Michael Movshev1 Institute for Advanced Study Einstein Dr 1, Princeton, NJ,USA [email protected] Dedicated to Yu.I. Manin on his 70-th birthday Summary. We construct a real-analytic CR supermanifold R, holomorphically em- bedded into a superquadric Q ⊂ P3j3 × P∗3j3. A CR distribution F on R enables • ¯ us to define a tangential CR complex (ΩF ; @). ¯ • We define a @-closed trace functional : ΩF ! ¡ and conjecture that a Chern- • ¯ Simons theory associated with a triple (ΩF ⊗ Matn; @; ) is equivalent to N=3, D=4 Yang-Mills theory with a gauge group U(n). We give some evidences to this conjecture. 1.1 Introduction Twistor methods in gauge theory have a long history (summarized in ref. [P90]). A common feature of these methods is that spacetime is replaced by a twistor (or ambitwistor) analytic manifold T . Equations of motion "emerge" (in terminology of Penrose) from complex geometry of T. The twistor approach turns to be a very useful technical innovation. For example difficult questions of classical gauge theory, e.g. the ones that appear in the theory of instantons, admit a translation into a considerably more simple questions of analytic geometry of space T. On this way classifications theorems in the theory of instantons has been obtained (see ref. [AHDM]). The quantum theory did not have a simple reformulation in the language of geometry of the space T so far. One of the reasons is that the quantum theory formulated formally in terms of a path integral requires a Lagrangian. Classical theory, as it was mentioned earlier, provides only equations of motion whose definition needs no metric.
    [Show full text]
  • Chapter 2: Linear Algebra User's Manual
    Preprint typeset in JHEP style - HYPER VERSION Chapter 2: Linear Algebra User's Manual Gregory W. Moore Abstract: An overview of some of the finer points of linear algebra usually omitted in physics courses. April 27, 2018 -TOC- Contents 1. Introduction 5 2. Basic Definitions Of Algebraic Structures: Rings, Fields, Modules, Vec- tor Spaces, And Algebras 6 2.1 Rings 6 2.2 Fields 6 2.3 Modules 8 2.4 Vector Spaces 9 2.5 Algebras 9 3. Linear Transformations 13 4. Basis And Dimension 15 4.1 Linear Independence 15 4.2 Free Modules 16 4.3 Vector Spaces 16 4.4 Linear Operators And Matrices 19 4.5 Determinant And Trace 22 5. New Vector Spaces from Old Ones 23 5.1 Direct sum 23 5.2 Quotient Space 27 5.3 Tensor Product 28 5.4 Dual Space 32 6. Tensor spaces 35 6.1 Totally Symmetric And Antisymmetric Tensors 36 6.2 Algebraic structures associated with tensors 38 6.2.1 An Approach To Noncommutative Geometry 41 7. Kernel, Image, and Cokernel 42 7.1 The index of a linear operator 43 8. A Taste of Homological Algebra 44 8.1 The Euler-Poincar´eprinciple 45 8.2 Chain maps and chain homotopies 46 8.3 Exact sequences of complexes 46 8.4 Left- and right-exactness 47 { 1 { 9. Relations Between Real, Complex, And Quaternionic Vector Spaces 47 9.1 Complex structure on a real vector space 47 9.2 Real Structure On A Complex Vector Space 50 9.2.1 Complex Conjugate Of A Complex Vector Space 52 9.2.2 Complexification 53 9.3 The Quaternions 55 9.4 Quaternionic Structure On A Real Vector Space 64 9.5 Quaternionic Structure On Complex Vector Space 65 9.5.1 Complex Structure On Quaternionic Vector Space 65 9.5.2 Summary 66 9.6 Spaces Of Real, Complex, Quaternionic Structures 66 10.
    [Show full text]