DOCSLIB.ORG

MATH 205B NOTES 1. 1/10 the Topic of the Course Is Functional Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
MATH 205B NOTES 1. 1/10 the Topic of the Course Is Functional Analysis MATH 205B NOTES MOOR XU NOTES FROM A COURSE BY RAFE MAZZEO Abstract. These notes were taken during Math 205B (Real Analysis) taught by Rafe Mazzeo in Winter 2012 at Stanford University. They were live-TEXed during lectures in vim and compiled using latexmk. Each lecture gets its own section. The notes are not edited afterward, so there may be typos; please email corrections to [email protected]. 1. 1/10 The topic of the course is functional analysis and applications. These include harmonic analysis and partial differential equations. These applications were the reason much of the theory was created. Today we will review some facts about topological spaces and metric spaces, and state some fundamental theorems. Definition 1.1. We have metric spaces (M; d). Here M is some set, and d : M × M ! R+ satisfying (1) d(x; y) ≥ 0 (2) d(x; y) = d(y; x) (3) d(x; y) = 0 iff x = y (4) d(x; y) ≤ d(x; z) + d(z; y). The simplest examples are things we know: n pP 2 Example 1.2. Euclidean space R with d(x; y) = (xi − yi) Other examples include the following: Example 1.3. • Finite sets with metrics. Here M is a finite set. • On the interval a ≤ x ≤ b, the real-valued continuous functions are C([a; b]). This has a metric d(f; g) = supa≤x≤b jf(x) − g(x)j. We can define B(f; ") as the set of all g such that d(f; g) < ". These are all continuous functions that never stray too far from f, inside a tube around f. • C1([a; b]) are continuously differentiable functions, so f 2 C1 if f 2 C0 and f 0 2 C0. This means that there are no sharp corners, and the derivative doesn't stray too far 0 0 either. The metric is d(f; g) = supx2[a;b](jf(x) − g(x)j + jf (x) − g (x)j). • L2[a; b] are functions with metric d(f; g) = (R jf(x) − g(x)j2 dx)1=2. Here, d(f; g) = 0 iff f = g almost everywhere; this notion is one reason why the Lebesgue integral exists. In fact, L2 functions are actually equivalence classes of functions, where f ∼ g if they agree almost everywhere. This eliminates some interesting functions. 1 p • L [a; b] for 1 ≤ p ≤ 1. Here kfk1 = sup jfj. The triangle inequality holds by the Holder inequality. All of these are special cases of normed spaces (V; k·k). Here, M is the vector space V , and the norm is k·k : V ! R+. By definition, we can write d(v; w) = kv − wk. A consequence is that d(v + b; w + b) = d(v; w). All of linear functional analysis is the interaction between the vector space and the underlying topology. Slightly more generally, we can talk about topological vector spaces. We want to apply our finite dimensional intuition, and we can for Hilbert spaces, but not for more exotic cases. But we don't have compactness, even for closures of balls! We'll start with the simplest examples and build up to more generality. At the bottom, we have Hilbert spaces (i.e. having inner products, like L2). More generally, there are Banach spaces (having norms such as Lp or Ck), but they are very ubiquitous but harder to deal with. Then there are Frechet spaces (with a countable set of norms like C1), and then locally convex topological vector spaces (e.g. distributions). Let's go back to general metric spaces. The really important property is the notion of completeness. Definition 1.4. A sequence of elements fxng in M is a Cauchy sequence if given any " > 0 there exists N such that m; n > N then d(xm; xn) < ". Definition 1.5. A space (M; d) is called complete if every Cauchy sequence has a limit in M. Example 1.6. Here are examples of complete spaces. • R or Rn are complete. The rational numbers are not complete, but its completion is all of R. But this makes many elements of R somewhat intangible { all we know is that they are limits of Cauchy sequences. • C0(I) or any Ck(I). • L1(I) is complete. (Riesz-Fisher Theorem) Example 1.7. Here are some counterexamples. 1 1 Consider (C ; k·kC0 ). That's certainly a normed metric space because C is a subset of C0, but it is not complete; there are differentiable functions that approach a sharp corner. The sup norm doesn't force the derivative to converge. Theorem 1.8. C0([a; b]) is complete. Proof. Let ffng be Cauchy with this norm. We want to produce a function which is the limit, and we need to show that it is still in C0. If x 2 I then ffn(x)g is a sequence in R, and it is still Cauchy, because jfn(x) − fm(x)j ≤ kfn − fmkC0 ! 0. So define f(x) = limn!1 fn(x). So now we've shown that this sequence ffng converges pointwise. Now, we need to show that f is C0, i.e. if " > 0 is fixed, then there exists δ > 0 such that we want jf(x) − f(y)j < " if jx − yj < δ. (This is actually uniform continuity, but we're working on a compact interval, so that's ok.) To show this, we have jf(x) − f(y)j ≤ jf(x) − fN (x)j + jfN (x) − fN (y)j + jfN (y) − f(y)j. Fix " > 0 and choose N such that kfn − fN k < " if n ≥ N. Then supx jf(x) − fN (x)j = supx limn!1 jfn(x) − fN (x)j ≤ supx supn≥N jfn(x) − fN (x)j = supn≥N kfn − fN k < ". Now the result follows by choosing N appropriately. 2 Now, we will head toward the main theorem that we will do today: the Arzela-Ascoli theorem. The question is: What sets of continuous functions are compact? Definition 1.9. Suppose that F ⊂ C0(I) is some collection of continuous functions. This is called uniformly equicontinuous if given any " > 0 there exists δ such that if jx − yj < δ and any f 2 F we have jf(x) − f(y)j < ". There are a lot of natural examples of equicontinuous functions, and this is the main motivating example. 1 0 Example 1.10. Let F = ff 2 C (I): kfkC1 ≤ Ag. Then F ⊂ C is uniformly equicontin- uous. This is because Z x Z y 0 0 jf(x) − f(y)j = f (t) dt ≤ jf (t)j dt ≤ Ajy − xj: y x Here are a couple of preparatory theorems. 0 Theorem 1.11. Suppose ffn 2 C (X; Y )g which is equicontinuous. If ffng converges point- wise to f, then f is continuous. Proof. Left to your vivid imaginations. Theorem 1.12. Let Y be complete, and let D ⊂ X be a dense set in X. Suppose that fn is equicontinuous on X and fn converges pointwise on D. Then, in fact, fn ! f uniformly. (Uniform needs that X is compact.) Proof. First, if x 2 D, we need to define f(x). Take any sequence xk ! x with xk 2 D. Then the limit limn!1 fn(xk) is well-defined. Define f(x) = lim lim fn(xk): k!1 n!1 There's something to check: any other sequence yk ! x gives the same limit. This comes from equicontinuity. We have d(xk; yl) < δ, so therefore ρ(fn(xk) − fn(yl)) < ", and this is well-defined. We still need to show that f is actually continuous, and the convergence is uniform. Go through the details yourself. We illustrate this with an example: Theorem 1.13. Suppose that ffng is a uniformly equicontinuous family on [0; 1] and D ⊂ [0; 1] is dense with pointwise convergence of fn on D. Then fn ! f uniformly. Proof. Fix ". There exists δ such that jfn(x) − fn(y)j < " if jx − yj < δ. For that δ, choose y1; : : : ; ym 2 [0; 1] such that yj 2 D and B(yj; δ) cover I. Then choose N so large that jfn(yi) − f(yi)j < " if n ≥ N. Now, jf(x)−f(~x)j ≤ jf(x)−fN (x)j+jfN (x)−fN (yj)j+jfN (yj)−fN (~x)j+jfN (~x)−f(~x)j. Theorem 1.14 (Arzela-Ascoli Theorem). Suppose that ffng is a family with fn : [0; 1] ! R uniformly bounded (i.e. there exists A such that jfn(x)j ≤ A for all n, x. Suppose that fn is equicontinuous. Then there exists fnj which converges uniformly. This is extremely useful and is the template for all compactness theorems in infinite dimensional spaces. The proof uses the diagonalization argument. 3 Lemma 1.15. Suppose that ffn(m)g is bounded in m and n uniformly. Then there exists nj such that fnj (m) converges for each m. Proof. For m = 1, fn(1) is a bounded sequence. Choose fn(1)j (1) convergent. Take a further subsequence such that fn(2)j (2) converges. So we get numbers n(k)j. Then define nj = n(j)j. This is the diagonalization. Proof of Theorem 1.14. Let D = Q \ I be dense. Call D = fxkg. For every rational, we have fn(xk) is uniformly bounded in n; k, so we can apply the diagonalization lemma above to choose nj ! 1 so fnj (xk) ! f(xk) for all xk. Now, using Theorem 1.12 gives the desired result. Remark. This is in fact a very general theorem about maps between general metric spaces.
Load more

Recommended publications
  • Tempered Distributions and the Fourier Transform
    CHAPTER 1 Tempered distributions and the Fourier transform Microlocal analysis is a geometric theory of distributions, or a theory of geomet- ric distributions. Rather than study general distributions { which are like general continuous functions but worse { we consider more specific types of distributions which actually arise in the study of differential and integral equations. Distri- butions are usually defined by duality, starting from very \good" test functions; correspondingly a general distribution is everywhere \bad". The conormal dis- tributions we shall study implicitly for a long time, and eventually explicitly, are usually good, but like (other) people have a few interesting faults, i.e. singulari- ties. These singularities are our principal target of study. Nevertheless we need the general framework of distribution theory to work in, so I will start with a brief in- troduction. This is designed either to remind you of what you already know or else to send you off to work it out. (As noted above, I suggest Friedlander's little book [4] - there is also a newer edition with Joshi as coauthor) as a good introduction to distributions.) Volume 1 of H¨ormander'streatise [8] has all that you would need; it is a good general reference. Proofs of some of the main theorems are outlined in the problems at the end of the chapter. 1.1. Schwartz test functions To fix matters at the beginning we shall work in the space of tempered distribu- tions. These are defined by duality from the space of Schwartz functions, also called the space of test functions of rapid decrease.
    [Show full text]
  • Lecture Notes: Introduction to Microlocal Analysis with Applications
    LECTURE NOTES: INTRODUCTION TO MICROLOCAL ANALYSIS WITH APPLICATIONS MIHAJLO CEKIC´ Contents Course Summary2 Notation 4 1. Recap: LCS topologies, distributions, Fourier transforms5 1.1. Topological vector spaces5 1.2. Space of distributions6 1.3. Schwartz kernel theorem8 1.4. Fourier transform9 2. Symbol classes, oscillatory integrals and Fourier Integral Operators 10 2.1. Symbol classes 10 2.2. Oscillatory integrals 13 3. Method of stationary phase 16 4. Algebra of pseudodifferential operators 18 4.1. Changes of variables (Kuranishi trick) 22 5. Elliptic operators and L2-continuity 23 5.1. L2-continuity 25 5.2. Sobolev spaces and PDOs 26 6. The wavefront set of a distribution 29 6.1. Operations with distributions: pullbacks and products 33 7. Manifolds 36 8. Quantum ergodicity 38 8.1. Ergodic theory 39 8.2. Statement and ingredients 41 8.3. Main proof of Quantum Ergodicity 44 9. Semiclassical calculus and Weyl's law 46 9.1. Proof of Weyl's law 48 References 50 1 2 M. CEKIC´ Course Summary Microlocal analysis studies singularities of distributions in phase space, by describing the behaviour of the singularity in both position and direction. It is a part of the field of partial differential equations, created by H¨ormander, Kohn, Nirenberg and others in 1960s and 1970s, and is used to study questions such as solvability, regularity and propagation of singularities of solutions of PDEs. To name a few other classical applications, it can be used to study asymptotics of eigenfunctions for elliptic operators, trace formulas and inverse problems. There have been recent exciting advances in the field and many applications to geometry and dynamical systems.
    [Show full text]
  • Characterizations of Continuous Operators on with the Strict Topology
    Tusi Mathematical Ann. Funct. Anal. (2021) 12:28 Research https://doi.org/10.1007/s43034-021-00112-1 Group ORIGINAL PAPER Characterizations of continuous operators on Cb(X) with the strict topology Marian Nowak1 · Juliusz Stochmal2 Received: 2 September 2020 / Accepted: 7 January 2021 / Published online: 16 February 2021 © The Author(s) 2021 Abstract C X Let X be a completely regular Hausdorf space and b( ) be the space of all bounded continuous functions on X, equipped with the strict topology . We study some ⋅ C X important classes of (, ‖ ‖E)-continuous linear operators from b( ) to a Banach E ⋅ space ( , ‖ ‖E) : -absolutely summing operators, compact operators and -nuclear operators. We characterize compact operators and -nuclear operators in terms of their representing measures. It is shown that dominated operators and -absolutely T C X → E summing operators ∶ b( ) coincide and if, in particular, E has the Radon– Nikodym property, then -absolutely summing operators and -nuclear operators coincide. We generalize the classical theorems of Pietsch, Tong and Uhl concern- ing the relationships between absolutely summing, dominated, nuclear and compact operators on the Banach space C(X), where X is a compact Hausdorf space. Keywords Spaces of bounded continuous functions · k-spaces · Radon vector measures · Strict topologies · Absolutely summing operators · Dominated operators · Nuclear operators · Compact operators · Generalized DF-spaces · Projective tensor product Mathematics Subject Classifcation 46G10 · 28A32 · 47B10 Communicated by Raymond Mortini. * Juliusz Stochmal [email protected] Marian Nowak [email protected] 1 Institute of Mathematics, University of Zielona Góra, ul. Szafrana 4A, 65-516 Zielona Gora, Poland 2 Institute of Mathematics, Kazimierz Wielki University, ul.
    [Show full text]
  • Arxiv:1503.04086V1 [Math-Ph] 13 Mar 2015 Schwartz Operators
    Schwartz operators M. Keyl TU München, Fakultät Mathematik, Boltzmannstr. 3, 85748 Garching, Germany [email protected] J. Kiukas School of Math. Sci., Univ. Nottingham, University Park, Nottingham, NG7 2RD, UK [email protected] R. F. Werner Inst. Theor. Physik, Leibniz Univ. Hannover, Appelstr. 2, 30167 Hannover, Germany [email protected] March 16, 2015 In this paper we introduce Schwartz operators as a non-commutative analog of Schwartz functions and provide a detailed discussion of their properties. We equip them in particular with a number of different (but equivalent) families of seminorms which turns the space of Schwartz operators into a Frechet space. The study of the topological dual leads to non-commutative tempered distributions which are discussed in detail as well. We show in particular that the latter can be identified with a certain class of quadratic forms, therefore making operations like products with bounded (and also some unbounded) operators and quantum harmonic analysis available to objects which are otherwise too singular for being a Hilbert space operator. Finally we show how the new methods can be applied by studying operator moment problems and convergence properties of fluctuation operators. 1 Introduction arXiv:1503.04086v1 [math-ph] 13 Mar 2015 The theory of tempered distributions is used extensively in various areas of mathematical physics, in order to regularise singular objects, most notably "delta-functions" that often appear as a result of some convenient idealisation (e.g. plane wave, or point interaction). The well-known intuitive idea is to make sense of the functional R2N φ(f) = S f(x)φ(x)dx, f ∈ S( ) (1) 1 to cases where φ is no longer a function, by making use of the highly regular behaviour of the 2N Schwartz functions f ∈ S(R ).
    [Show full text]
  • Lecture 15 1 Nov, 2016
    18.155 LECTURE 15 1 NOV, 2016 RICHARD MELROSE Abstract. Notes before and after lecture. 1. Before lecture • Schwartz' kernel theorem. • Smoothing operators • Locality (Peetre's Theorem) • Proper supports • Pseudolocality 2. After lecture • Fixed the confusion between φ and in the notes below (I hope). 1 −∞ 0 • I also described the transpose of an operator L : Cc (Ω) −! C (Ω ) as t 1 0 −∞ t the operator L : Cc (Ω ) −! C (Ω) with kernel K (x; y) = K(y; x) if K is the kernel of L: Notice that it satisfies the identity (1) hL ; φi = h ; Ltφi in terms of the real pairing extending the integral. t −∞ • If L and L both restrict and extend to define operators Cc (Ω) −! 1 0 −∞ 0 1 t C (Ω ) and Cc (Ω ) −! C (Ω) then L (and of course L ) is called a smoothing operator. This is equivalent to the condition K 2 C1(Ω0 × Ω): Having talked about operators on Hilbert space we now want to consider more general operators. The most general type of linear map that arises here is from a `small' space { a space of test functions { to a `large' space { a space of distributions. To be concrete let's consider linear maps of two types 1 −∞ 0 n 0 N L : Cc (Ω) −! C (Ω ); Ω ⊂ R ; Ω ⊂ R open (2) n 0 N L : S(R ) −! S (R ): In fact we will come to an understanding of the first type through the second. So consider a linear map of the second type in (2). We need to make some assumptions of continuity.
    [Show full text]
  • Kernel Theorems in Coorbit Theory
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 6, Pages 346–364 (November 14, 2019) https://doi.org/10.1090/btran/42 KERNEL THEOREMS IN COORBIT THEORY PETER BALAZS, KARLHEINZ GROCHENIG,¨ AND MICHAEL SPECKBACHER Abstract. We prove general kernel theorems for operators acting between coorbit spaces. These are Banach spaces associated to an integrable repre- sentation of a locally compact group and contain most of the usual function spaces (Besov spaces, modulation spaces, etc.). A kernel theorem describes the form of every bounded operator between a coorbit space of test functions and distributions by means of a kernel in a coorbit space associated to the tensor product representation. As special cases we recover Feichtinger’s ker- nel theorem for modulation spaces and the recent generalizations by Cordero and Nicola. We also obtain a kernel theorem for operators between the Besov ˙ 0 ˙ 0 spaces B1,1 and B∞,∞. 1. Introduction Kernel theorems assert that every “reasonable” operator can be written as a “generalized” integral operator. For instance, the Schwartz kernel theorem states that a continuous linear operator A : S(Rd) →S(Rd) possesses a unique distribu- tional kernel K ∈S(R2d) such that (1) Af, g = K, g ⊗ f,f,g∈S(Rd) . If K is a locally integrable function, then Af, g = K(x, y)f(y)g(x)dydx, f, g ∈S(Rd), Rd and thus A has indeed the form of an integral operator. Similar kernel theorems hold for continuous operators from D(Rd) →D(Rd) [24, Theorem 5.2] and for Gelfand-Shilov spaces and their distribution spaces [21].
    [Show full text]
  • On the Laplacian on a Space of White Noise Functionals
    TSUKUBA J. MATH. Vol. 19 No. 1 (1995), 93―119 ON THE LAPLACIAN ON A SPACE OF WHITE NOISE FUNCTIONALS By Isamu Doku* §1. Introduction. We are greatly interested in the Laplacian on a space of white noise func- tionals. To have in mind aspects of application to mathematical physics, we can say that it is common in general to use the weak derivative Dona given basic Hilbert space, so as to define dv which just corresponds to the de Rham exterior differentialoperator. In doing so, one of the remarkable characteristics of our work consistsin adoption of the Hida differentialdt instead of D. This distinction from other related works does provide a framework of analysis equipped with the function for perception of the time t, with the result that it is converted into a more flexible and charming theory which enables us to treat time evolution directly. It can be said, therefore, that our work is suc- cessfulin deepening works about the general theory done by Arai-Mitoma [2], not only on a qualitative basis but also from the applicatory point of view in direct description of operators in terms of time evolution. The differentialdt has its adjoint operator df in Hida sense and it is called the Kubo operator. Indeed, df is realized by extending the functional space even into the widest one (E)*, where a Gelfand triple(E) C, (L2) Q (E)* is a fundamental setting in white noise analysis, in accordance with our more general choice of the basic Hilbert space H. On the contrary, we define the adjoint operator dp* of dp associated with dt without extending the space up to that much.
    [Show full text]
  • Invariant Hilbert Subspaces of the Oscillator Representation Aparicio, S
    Invariant Hilbert subspaces of the oscillator representation Aparicio, S. Citation Aparicio, S. (2005, October 31). Invariant Hilbert subspaces of the oscillator representation. Retrieved from https://hdl.handle.net/1887/3507 Version: Corrected Publisher’s Version Licence agreement concerning inclusion of doctoral thesis License: in the Institutional Repository of the University of Leiden Downloaded from: https://hdl.handle.net/1887/3507 Note: To cite this publication please use the final published version (if applicable). Invariant Hilbert subspaces of the oscillator representation Proefschrift ter verkrijging van de graad van Doctor aan de Universiteit Leiden, op gezag van de Rector Magni¯cus Dr. D. D. Breimer, hoogleraar in de faculteit der Wiskunde en Natuurwetenschappen en die der Geneeskunde, volgens besluit van het College voor Promoties te verdedigen op maandag 31 oktober 2005 te klokke 15.15 uur door Sof¶³a Aparicio Secanellas geboren te Zaragoza (Spanje) op 22 juni 1977 Samenstelling van de promotiecommissie: promotor: Prof. dr. G. van Dijk referent: Dr. E. P. van den Ban (Universiteit Utrecht) overige leden: Prof. dr. E. G. F. Thomas (Rijksuniversiteit Groningen) Dr. M. F. E. de Jeu Prof. dr. M. N. Spijker Prof. dr. S. M. Verduyn Lunel A mi familia y a Guillermo Thomas Stieltjes Institute for Mathematics Contents Publication history 9 1 Introduction 11 1.1 Some basic de¯nitions . 11 1.2 Connection between representation theory and quantum mechanics 13 1.3 Overview of this thesis . 14 1.3.1 Representations of SL(2; IR) and SL(2; C) . 14 1.3.2 The metaplectic representation . 14 1.3.3 Theory of invariant Hilbert subspaces .
    [Show full text]
  • Native Banach Spaces for Splines and Variational Inverse Problems∗
    Native Banach spaces for splines and variational inverse problems∗ Michael Unser and Julien Fageoty April 25, 2019 Abstract We propose a systematic construction of native Banach spaces for general spline-admissible operators L. In short, the native space for L and the (dual) norm 0 is the largest space of functions f : d k · kX R R such that Lf 0 < , subject to the constraint that the growth-restricted! nullk spacekX of1L be finite-dimensional. This space, denoted by L0 , is specified as the dual of the pre-native space L, which is itself obtainedX through a suitable completion process. TheX main difference with prior constructions (e.g., reproducing kernel Hilbert spaces) is that our approach involves test functions rather than sums of atoms (e.g., kernels), which makes it applicable to a much broader class of norms, including total variation. Under specific admissibility and compatibility hypotheses, we lay out the direct-sum topology of L and L0 , and identify the whole family of equivalent norms. Our constructionX X ensures that the native space and its pre-dual are endowed with a fundamental Schwartz-Banach property. In practical terms, this means that L0 is rich enough to reproduce any function with an arbitrary degreeX of precision. Contents 1 Introduction 2 arXiv:1904.10818v1 [math.FA] 24 Apr 2019 2 Preliminaries 6 2.1 Schwartz-Banach spaces . .6 2.2 Characterization of linear operators and their adjoint . 10 ∗The research leading to these results has received funding from the Swiss National Science Foundation under Grant 200020-162343. yBiomedical Imaging Group, École polytechnique fédérale de Lausanne (EPFL), Sta- tion 17, CH-1015, Lausanne, Switzerland ([email protected]).
    [Show full text]
  • Banach Gelfand Triple: a Soft Introduction to Mild Distributions
    Goal SO-BGTr The function space Zoo BUPUs TF-Analysis TILS Role of SO Kernel Theorem Gabor Multipliers Just one Fourier Transform! Mutual Approximations 2 Conclusion References Numerical Harmonic Analysis Group Banach Gelfand Triple: A Soft Introduction to Mild Distributions Hans G. Feichtinger, Univ. Vienna & Charles Univ. Prague [email protected] www.nuhag.eu Peoples Friendship University of Russia, Moscow, Russia . Theory of Functions and Functional Spaces, Nov. 2018 Banach Gelfand Triple:A Soft Introduction to Mild Distributions Hans G. Feichtinger, Univ. Vienna & Charles Univ. Prague [email protected] www.nuhag.eu Goal SO-BGTr The function space Zoo BUPUs TF-Analysis TILS Role of SO Kernel Theorem Gabor Multipliers Just one Fourier Transform! Mutual Approximations 2 Conclusion References Goal of the Talk This talk is to be understood as a contribution to the theory of function and functional (meaning generalized function) spaces. This theory has a long history and not only experts find a rich family of such spaces, including Banach spaces, but also topological vector spaces (in particular nuclear Frechet spaces) among them. Among the Banach spaces certainly the Lebesgue Spaces p d L (R ); k · kp are the most fundamental one. Their study has definitely shaped the development of early (linear) functional analysis, with case p = 2 being the prototype of a Hilbert spaces. d 0 d The Schwartz space S(R ) and it dual, the space S (R ) of tempered distributions being the most prominent among them, r but there are also the Gelfand-Shilov classes Ss providing the basis for the theory of ultra-distributions.
    [Show full text]
  • Schwartz' Kernel Theorem, Tensor Products, Nuclearity
    (May 7, 2020) 10a. Schwartz kernel theorems, tensor products, nuclearity Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http:=/www.math.umn.edu/egarrett/m/real/notes 2019-20/10a Schwartz kernel theorem.pdf] 1. Concrete Schwartz' kernel theorems: statements 2. Examples of Schwartz kernels 3. Adjunction of Hom and ⊗ 4. Topologies on Hom(Y; Z) 5. Continuity conditions on bilinear maps 6. Some ambiguity removed 7. Nuclear Fr´echet spaces 8. Adjunction: Schwartz kernel theorem for nuclear Fr´echet spaces 9. D(Tn) is nuclear Fr´echet 10. D(Tm) ⊗ D(Tn) ≈ D(Tm+n) 11. Schwartz kernel theorem for D(Tn) 12. Nuclear LF-spaces 13. Adjunction: Schwartz kernel theorem for nuclear LF-spaces 14. D(Rn) is a nuclear LF-space 15. D(Rm) ⊗ D(Rn) ≈ D(Rm+n) 16. Appendix: joint continuity of bilinear maps 17. Appendix: convex hulls 18. Appendix: Hilbert-Schmidt operators 19. Appendix: non-existence of tensor products of infinite-dimensional Hilbert spaces Hilbert-Schmidt operators T : L2(Rm) ! L2(Rn) are exactly those continuous linear operators given by kernels K(x; y) by [1] Z T f(y) = K(x; y) f(x) dx Rm Here K(x; y) is a Schwartz kernels K(x; y) 2 L2(Rm+n). But most continuous linear maps T : L2(Rm) ! L2(Rn) are not Hilbert-Schmidt, so do not have Schwartz kernels in L2(Rm+n). The obstacle is not just the non-compactness of R, as most continuous T : L2(Tm) ! L2(Tn) do not have kernels in this sense, either.
    [Show full text]
  • Index Theory of Uniform Pseudodifferential Operators
    Index theory of uniform pseudodifferential operators Alexander Engel Fakultät für Mathematik Universität Regensburg 93040 Regensburg, GERMANY [email protected] Abstract We generalize Roe’s index theorem for graded generalized Dirac operators on amenable manifolds to multigraded elliptic uniform pseudodifferential operators. This generalization will follow from a local index theorem that is valid on any man- ifold of bounded geometry. This local formula incorporates the uniform estimates that are present in the definition of our class of pseudodifferential operators. We will revisit Špakula’s uniform K-homology and show that multigraded elliptic uniform pseudodifferential operators naturally define classes in it. For this we will investigate uniform K-homology more closely, e.g., construct the external product and show invariance under weak homotopies. The latter will be used to refine and extend Špakula’s results about the rough Baum–Connes assembly map. We will identify the dual theory of uniform K-homology. We will give a simple definition of uniform K-theory for all metric spaces and in the case of manifolds of bounded geometry we will give an interpretation of it via vector bundles of bounded geometry. Using a version of Mayer–Vietoris induction that is adapted to our needs, we will prove Poincaré duality between uniform K-theory and uniform K-homology for spinc manifolds of bounded geometry. We will construct Chern characters from uniform K-theory to bounded de Rham cohomology and from uniform K-homology to uniform de Rham homology. Using the adapted Mayer–Vietoris induction we will also show that these Chern characters induce isomorphisms after a certain completion that also kills torsion.
    [Show full text]