DOCSLIB.ORG

Playing with Elementary Functional Integrals & Contours + Imagery

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Playing with Elementary Functional Integrals & Contours + Imagery A Note: Playing With Elementary Functional Integrals & Contours + Imagery John Gill Summer 2017 Abstract: Starting from a rudimentary definition of functional integral and basic complex variable theory several examples of functional integrals and their values are derived. In addition, images of the complex plane in which each point is associated with a functional integral are presented. This is not the functional integral used in physics, nor are these examples related to applications. From Wikipedia: “ In an ordinary integral there is a function to be integrated (the integrand) and a region of space over which to integrate the function (the domain of integration). The process of integration consists of adding up the values of the integrand for each point of the domain of integration . where the domain of integration is divided into smaller and smaller regions. For each small region, the value of the integrand cannot vary much, so it may be replaced by a single value. In a functional integral the domain of integration is a space of functions. For each function, the integrand returns a value to add up.” b n The common integral fxdx()= Lim∑ fc ()kn, ∆ x kn , , where x is the (real) variable of ∫ n→∞ a k=1 n integration and f( x ) is the integrand. Π[a , b ] = x is a partition of [a , b ] and n{ k, n }k=1 ∆xkn, = x k+ 1, n − x kn , with xkn,≤ c kn , ≤ x k+ 1, n . All the subintervals shrink to 0 as n → ∞ . On the other hand, the integral ∫F[ f ] dµ T has as its variable of integration “points” in a S function space S , i.e., functions f∈ S . And the “subintervals” are sets (σ − algebras ) of functions requiring a non-negative definition of “size” or “measure”, µT , paralleling the Riemann concept. F[ f ] is a functional operating on a function f∈ S and returning a real or complex number for each such function. I start from this very rudimentary and simple definition of functional integral and not from the usual interpretation seen in path integrals. Also, when measures are discussed I include pre- measures and cylinder measures under this rubric. The space S will be a space of defined contours in the complex plane. In order to establish a functional integral in the context of sets of contours, S , we need a way of quantifying or measuring sets, T , within S , and a designation of a collection of such sets dependent upon n; also, a reasonable way of extracting an individual contour, z( t ) , within each set and assigning a numerical value to it. We want the “size” of each of these sets to diminish to 0 as n → ∞ . Thus the “functional integral” here is a hybrid concept. Example : S :{ztzttit():()=+ρ , 0 ≤≤ ρ 1, t ∈ [0,1] } . Think of these contours as vectors in the first quadrant. Assume the unit square is divided into n× n squares. Focus on the right-most k k +1 vertical line, divided into equal subintervals. Let Tk, n ={(): z t n ≤ρ < n } . The simplest measure 1 2 k is µT = . Now, define a functional as Fzt[ ( )]= 1 + i ρ , ρ = , where the contour k, n n kn, kn , kn , n k k +1 terminates at lowest point of the subinterval [n , n ) . Then n k2 1 F[ z ]µ T= Lim∑ 1 +⋅≈ .5417, n = 10,000 ∫ n→∞ 2 S k=1 n n This particular contour from Tk, n is representative of all contours in this set since n n n 2 2k − 1 11+−ρ 2 1 1 +≤k →→∞0, n n∑()k, n n ∑()n2 ∑ 3 k=1 k = 1 k = 1 n Of course, this is a Riemann integral in disguise. Let the set Sz be a space of contours ζ s (t ) with t∈[0,1], s ∈ [0,1] . s denotes a family of individual contours originating at the point z , and t ranging from 0 to 1 describes a contour. The first choice of a contour space is Sz:{ζ s ( t ): ζ s ( t )=+ z sSin (5 t ) − itCos (5 s ), z =+1 i } A collection of sets in this space is given by Tab(,)={ζ s ():0 t ≤≤<< asb 1 } . These cylinder sets generate a σ −algebra . The measure we apply is the distance traversed by the points ζ s(1) over the s -intervals (the red and green contour in the image below). The next step is to designate a particular collection of sets k k +1 Tk, n={ζ s (): tsn ≤< n } , kn:0 → , n → ∞ . k+1 n 2 Then µTk, n= ∫ d ζ s (1) . For a functional on Sz , set Fζk= z − ζ k (1) for that particular n n k n ⇒ contour from each subset. Then λ(z )= ∫ FdT [ ζµs ] λ(1+ i ) = ∫ FdT[ ζµs ] ≈ 2.2078 Sz S 1+i An accurate representation would not show individual contours as s is a continuous variable. z(1+ t ( e s − 1) ) Example : Let ζ ()t= , zxiy = + . In the image below z = 1+i. The measure s 1 −t2 () xs + iy and functional are the same. Image over a region: Dark = small moduli , light = large. λ(z ) = ∫ F[ ζs ] dT µ : Sz [-10,10] z Example : ζ (t )= , z = x + iy . The same details as in the previous s 1+t() Cos () xs − iSin () ys example. z=1+I in the first image. λ(z ) = ∫ F[ ζs ] dT µ : The image is over [-10,10]. Sz z(1+ iSin () t Cos ( sxy ) ) Example : ζ s (t )= , z = x + iy z=1+I λ(z ) = ∫ F[ ζs ] dT µ : 1+t() Cos () xs − iSin () ys Sz [-10,10] z Example : ζ s (t )= , z = x + iy , z=1+I, λ(z ) = ∫ F[ ζs ] dT µ : 1+t() sCos (5 x ) + iSin (5 y ) Sz [-10,10] z( Cos(2 t )+ iSin (2 t ) ) Example : ζ s (t )= , z = x + iy , z=1+I λ(z ) = ∫ d µ T : 1+ts() Cos () xy + iSin () xy Sz [-10,10] A contour area measure : Example : ζ s()2t= t + is ( + 1) Sin (2) st . Let Tab(,)={ζ s ():0 t ≤≤<< asb 1 } . This cylinder set generates a sigma-algebra. For a measure, set µTab( , )= Area between ζa( t ) and ζ b ( t ) . And k k +1 for a functional, set F[ζb ( t )]= ζ b (1) . Specify Tk, n={ζ s ( t ): n ≤ s < n } . Then n F[ζµs ] dT = LimF∑ [ ζµ kn/ (1)] T kn , ≈ 3.2455 . ∫ n→∞ k=0 S0 nk+1 n k With µTk, n =+1() 1 − Cos (2) −+− 1() 1 Cos (2) k+ 1 n k n Contours are related to complex vector fields in the following way: Given a vector field f( z ) in C , to determine a formula for a contour, z( t ) , write dz =ϕ()z = fz () − z and solve for z( t ) . For instance, ϕ()zz= 2 ⇒ fzzz() = 2 + , leading to dt z z( t ) = 0 . 1− z0 t This approach is taken in the following example where the focus is upon an attractor in a complex vector field: Example : Consider the set of contours in the vector space f( z ) = z 2 . There are two fixed points in the space, α = 0 (attractor) and β = 1 (repulsor), that govern flow. Each circle centered at α = 0 is a locus of terminating points of contours z( t ), t ∈[0,1] . Corresponding to * each such circle of radius rj (in red in figure) is a circle of radius rj (in blue) of initial points of 1 these contours. Let S be the space of contours of the form {z( t ): z (1) ≤ 2} . k Now set ε =, 1 ≤k ≤ n . For the rings R about the origin, formed by consecutive radii k, n 2n k, n εkn,, ε k+ 1, n , set Tkn,={ ztz ( ): (1) ∈ R kn , } . We have dz z 1 ⇒ 0 =ϕ(z ) = fzz ( ) − zt( ) = t t . Partition D= z ≤ into diminishing rings dt zee0(1− ) + 2 n D= Lim∪ Rknkn,,, R =≤ zε kn ,∩ z > ε kn+ 1, . Now observe that n→∞ () () k=1 Tkn,={ ztz( ): (1) ∈ R kn , } corresponds in a one-to-one manner to Zkn,={ zz 0: (1) ∈ R kn , } . The two blue circles in the figure define the boundary of Zk, n . An obvious choice for µTk, n would be the area between the blue circles. However, the algebra involved is unpleasant, so, having no application in mind, we default to the much simpler difference between the two circumferences: ck22+ ck 2 − 4 n 2 ∆C = 4 eπ n , c= e − 1 k, n (ck22−4 nck 22 )( ( + 1) 2 − 4 n 2 ) The remaining decisions are to designate a zkn,( t ) ∈ T kn , as n → ∞ and F[ z ( t )] in a way that n ∑Fz[kn, ( t )] ⋅ µ T kn , is a proper mathematical object and is computable. Toward this end k=1 observe that very large values of n produce z0 ' s virtually on the circumference of a (blue) circle Ck, n . So that all zkn,( t ) ∈ T kn , roughly correspond to points on this one circle. Hence, we choose 4eπ nk Fz[ ( t )] = C = kn, kn , ()c2 k 2− 4 n 2 Define n n 22 2 2 2 2 2 kck(+ ck − 4 n ) FzdT[ ]µ= Lim∑ Fz [kn, ( t )] ⋅ µ T kn , = 16 e π nLim ∑ 2 . ∫ n→∞ n →∞ 22 2 2 2 2 S k=1 k = 1 ()()ck−4 n ck ( + 1) − 4 n And ∫FzdT[ ]µ ≈ 44.324, n = 100,000 . S 1 Example : The same as the one above, except ε =, 2 ≤ k →∞ .
Load more

Recommended publications
  • FUNCTIONAL ANALYSIS 1. Banach and Hilbert Spaces in What
    FUNCTIONAL ANALYSIS PIOTR HAJLASZ 1. Banach and Hilbert spaces In what follows K will denote R of C. Definition. A normed space is a pair (X, k · k), where X is a linear space over K and k · k : X → [0, ∞) is a function, called a norm, such that (1) kx + yk ≤ kxk + kyk for all x, y ∈ X; (2) kαxk = |α|kxk for all x ∈ X and α ∈ K; (3) kxk = 0 if and only if x = 0. Since kx − yk ≤ kx − zk + kz − yk for all x, y, z ∈ X, d(x, y) = kx − yk defines a metric in a normed space. In what follows normed paces will always be regarded as metric spaces with respect to the metric d. A normed space is called a Banach space if it is complete with respect to the metric d. Definition. Let X be a linear space over K (=R or C). The inner product (scalar product) is a function h·, ·i : X × X → K such that (1) hx, xi ≥ 0; (2) hx, xi = 0 if and only if x = 0; (3) hαx, yi = αhx, yi; (4) hx1 + x2, yi = hx1, yi + hx2, yi; (5) hx, yi = hy, xi, for all x, x1, x2, y ∈ X and all α ∈ K. As an obvious corollary we obtain hx, y1 + y2i = hx, y1i + hx, y2i, hx, αyi = αhx, yi , Date: February 12, 2009. 1 2 PIOTR HAJLASZ for all x, y1, y2 ∈ X and α ∈ K. For a space with an inner product we define kxk = phx, xi . Lemma 1.1 (Schwarz inequality).
    [Show full text]
  • Definition of Riemann Integrals
    The Definition and Existence of the Integral Definition of Riemann Integrals Definition (6.1) Let [a; b] be a given interval. By a partition P of [a; b] we mean a finite set of points x0; x1;:::; xn where a = x0 ≤ x1 ≤ · · · ≤ xn−1 ≤ xn = b We write ∆xi = xi − xi−1 for i = 1;::: n. The Definition and Existence of the Integral Definition of Riemann Integrals Definition (6.1) Now suppose f is a bounded real function defined on [a; b]. Corresponding to each partition P of [a; b] we put Mi = sup f (x)(xi−1 ≤ x ≤ xi ) mi = inf f (x)(xi−1 ≤ x ≤ xi ) k X U(P; f ) = Mi ∆xi i=1 k X L(P; f ) = mi ∆xi i=1 The Definition and Existence of the Integral Definition of Riemann Integrals Definition (6.1) We put Z b fdx = inf U(P; f ) a and we call this the upper Riemann integral of f . We also put Z b fdx = sup L(P; f ) a and we call this the lower Riemann integral of f . The Definition and Existence of the Integral Definition of Riemann Integrals Definition (6.1) If the upper and lower Riemann integrals are equal then we say f is Riemann-integrable on [a; b] and we write f 2 R. We denote the common value, which we call the Riemann integral of f on [a; b] as Z b Z b fdx or f (x)dx a a The Definition and Existence of the Integral Left and Right Riemann Integrals If f is bounded then there exists two numbers m and M such that m ≤ f (x) ≤ M if (a ≤ x ≤ b) Hence for every partition P we have m(b − a) ≤ L(P; f ) ≤ U(P; f ) ≤ M(b − a) and so L(P; f ) and U(P; f ) both form bounded sets (as P ranges over partitions).
    [Show full text]
  • Generalizations of the Riemann Integral: an Investigation of the Henstock Integral
    Generalizations of the Riemann Integral: An Investigation of the Henstock Integral Jonathan Wells May 15, 2011 Abstract The Henstock integral, a generalization of the Riemann integral that makes use of the δ-fine tagged partition, is studied. We first consider Lebesgue’s Criterion for Riemann Integrability, which states that a func- tion is Riemann integrable if and only if it is bounded and continuous almost everywhere, before investigating several theoretical shortcomings of the Riemann integral. Despite the inverse relationship between integra- tion and differentiation given by the Fundamental Theorem of Calculus, we find that not every derivative is Riemann integrable. We also find that the strong condition of uniform convergence must be applied to guarantee that the limit of a sequence of Riemann integrable functions remains in- tegrable. However, by slightly altering the way that tagged partitions are formed, we are able to construct a definition for the integral that allows for the integration of a much wider class of functions. We investigate sev- eral properties of this generalized Riemann integral. We also demonstrate that every derivative is Henstock integrable, and that the much looser requirements of the Monotone Convergence Theorem guarantee that the limit of a sequence of Henstock integrable functions is integrable. This paper is written without the use of Lebesgue measure theory. Acknowledgements I would like to thank Professor Patrick Keef and Professor Russell Gordon for their advice and guidance through this project. I would also like to acknowledge Kathryn Barich and Kailey Bolles for their assistance in the editing process. Introduction As the workhorse of modern analysis, the integral is without question one of the most familiar pieces of the calculus sequence.
    [Show full text]
  • Worked Examples on Using the Riemann Integral and the Fundamental of Calculus for Integration Over a Polygonal Element Sulaiman Abo Diab
    Worked Examples on Using the Riemann Integral and the Fundamental of Calculus for Integration over a Polygonal Element Sulaiman Abo Diab To cite this version: Sulaiman Abo Diab. Worked Examples on Using the Riemann Integral and the Fundamental of Calculus for Integration over a Polygonal Element. 2019. hal-02306578v1 HAL Id: hal-02306578 https://hal.archives-ouvertes.fr/hal-02306578v1 Preprint submitted on 6 Oct 2019 (v1), last revised 5 Nov 2019 (v2) 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. Worked Examples on Using the Riemann Integral and the Fundamental of Calculus for Integration over a Polygonal Element Sulaiman Abo Diab Faculty of Civil Engineering, Tishreen University, Lattakia, Syria [email protected] Abstracts: In this paper, the Riemann integral and the fundamental of calculus will be used to perform double integrals on polygonal domain surrounded by closed curves. In this context, the double integral with two variables over the domain is transformed into sequences of single integrals with one variable of its primitive. The sequence is arranged anti clockwise starting from the minimum value of the variable of integration. Finally, the integration over the closed curve of the domain is performed using only one variable.
    [Show full text]
  • Linear Spaces
    Chapter 2 Linear Spaces Contents FieldofScalars ........................................ ............. 2.2 VectorSpaces ........................................ .............. 2.3 Subspaces .......................................... .............. 2.5 Sumofsubsets........................................ .............. 2.5 Linearcombinations..................................... .............. 2.6 Linearindependence................................... ................ 2.7 BasisandDimension ..................................... ............. 2.7 Convexity ............................................ ............ 2.8 Normedlinearspaces ................................... ............... 2.9 The `p and Lp spaces ............................................. 2.10 Topologicalconcepts ................................... ............... 2.12 Opensets ............................................ ............ 2.13 Closedsets........................................... ............. 2.14 Boundedsets......................................... .............. 2.15 Convergence of sequences . ................... 2.16 Series .............................................. ............ 2.17 Cauchysequences .................................... ................ 2.18 Banachspaces....................................... ............... 2.19 Completesubsets ....................................... ............. 2.19 Transformations ...................................... ............... 2.21 Lineartransformations.................................. ................ 2.21
    [Show full text]
  • Lecture 15-16 : Riemann Integration Integration Is Concerned with the Problem of finding the Area of a Region Under a Curve
    1 Lecture 15-16 : Riemann Integration Integration is concerned with the problem of ¯nding the area of a region under a curve. Let us start with a simple problem : Find the area A of the region enclosed by a circle of radius r. For an arbitrary n, consider the n equal inscribed and superscibed triangles as shown in Figure 1. f(x) f(x) π 2 n O a b O a b Figure 1 Figure 2 Since A is between the total areas of the inscribed and superscribed triangles, we have nr2sin(¼=n)cos(¼=n) · A · nr2tan(¼=n): By sandwich theorem, A = ¼r2: We will use this idea to de¯ne and evaluate the area of the region under a graph of a function. Suppose f is a non-negative function de¯ned on the interval [a; b]: We ¯rst subdivide the interval into a ¯nite number of subintervals. Then we squeeze the area of the region under the graph of f between the areas of the inscribed and superscribed rectangles constructed over the subintervals as shown in Figure 2. If the total areas of the inscribed and superscribed rectangles converge to the same limit as we make the partition of [a; b] ¯ner and ¯ner then the area of the region under the graph of f can be de¯ned as this limit and f is said to be integrable. Let us de¯ne whatever has been explained above formally. The Riemann Integral Let [a; b] be a given interval. A partition P of [a; b] is a ¯nite set of points x0; x1; x2; : : : ; xn such that a = x0 · x1 · ¢ ¢ ¢ · xn¡1 · xn = b.
    [Show full text]
  • General Inner Product & Fourier Series
    General Inner Products 1 General Inner Product & Fourier Series Advanced Topics in Linear Algebra, Spring 2014 Cameron Braithwaite 1 General Inner Product The inner product is an algebraic operation that takes two vectors of equal length and com- putes a single number, a scalar. It introduces a geometric intuition for length and angles of vectors. The inner product is a generalization of the dot product which is the more familiar operation that's specific to the field of real numbers only. Euclidean space which is limited to 2 and 3 dimensions uses the dot product. The inner product is a structure that generalizes to vector spaces of any dimension. The utility of the inner product is very significant when proving theorems and runing computations. An important aspect that can be derived is the notion of convergence. Building on convergence we can move to represenations of functions, specifally periodic functions which show up frequently. The Fourier series is an expansion of periodic functions specific to sines and cosines. By the end of this paper we will be able to use a Fourier series to represent a wave function. To build toward this result many theorems are stated and only a few have proofs while some proofs are trivial and left for the reader to save time and space. Definition 1.11 Real Inner Product Let V be a real vector space and a; b 2 V . An inner product on V is a function h ; i : V x V ! R satisfying the following conditions: (a) hαa + α0b; ci = αha; ci + α0hb; ci (b) hc; αa + α0bi = αhc; ai + α0hc; bi (c) ha; bi = hb; ai (d) ha; aiis a positive real number for any a 6= 0 Definition 1.12 Complex Inner Product Let V be a vector space.
    [Show full text]
  • Function Space Tensor Decomposition and Its Application in Sports Analytics
    East Tennessee State University Digital Commons @ East Tennessee State University Electronic Theses and Dissertations Student Works 12-2019 Function Space Tensor Decomposition and its Application in Sports Analytics Justin Reising East Tennessee State University Follow this and additional works at: https://dc.etsu.edu/etd Part of the Applied Statistics Commons, Multivariate Analysis Commons, and the Other Applied Mathematics Commons Recommended Citation Reising, Justin, "Function Space Tensor Decomposition and its Application in Sports Analytics" (2019). Electronic Theses and Dissertations. Paper 3676. https://dc.etsu.edu/etd/3676 This Thesis - Open Access is brought to you for free and open access by the Student Works at Digital Commons @ East Tennessee State University. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of Digital Commons @ East Tennessee State University. For more information, please contact [email protected]. Function Space Tensor Decomposition and its Application in Sports Analytics A thesis presented to the faculty of the Department of Mathematics East Tennessee State University In partial fulfillment of the requirements for the degree Master of Science in Mathematical Sciences by Justin Reising December 2019 Jeff Knisley, Ph.D. Nicole Lewis, Ph.D. Michele Joyner, Ph.D. Keywords: Sports Analytics, PCA, Tensor Decomposition, Functional Analysis. ABSTRACT Function Space Tensor Decomposition and its Application in Sports Analytics by Justin Reising Recent advancements in sports information and technology systems have ushered in a new age of applications of both supervised and unsupervised analytical techniques in the sports domain. These automated systems capture large volumes of data points about competitors during live competition.
    [Show full text]
  • Using Functional Distance Measures When Calibrating Journey-To-Crime Distance Decay Algorithms
    USING FUNCTIONAL DISTANCE MEASURES WHEN CALIBRATING JOURNEY-TO-CRIME DISTANCE DECAY ALGORITHMS A Thesis Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Master of Natural Sciences in The Interdepartmental Program of Natural Sciences by Joshua David Kent B.S., Louisiana State University, 1994 December 2003 ACKNOWLEDGMENTS The work reported in this research was partially supported by the efforts of Dr. James Mitchell of the Louisiana Department of Transportation and Development - Information Technology section for donating portions of the Geographic Data Technology (GDT) Dynamap®-Transportation data set. Additional thanks are paid for the thirty-five residence of East Baton Rouge Parish who graciously supplied the travel data necessary for the successful completion of this study. The author also wishes to acknowledge the support expressed by Dr. Michael Leitner, Dr. Andrew Curtis, Mr. DeWitt Braud, and Dr. Frank Cartledge - their efforts helped to make this thesis possible. Finally, the author thanks his wonderful wife and supportive family for their encouragement and tolerance. ii TABLE OF CONTENTS ACKNOWLEDGMENTS .............................................................................................................. ii LIST OF TABLES.......................................................................................................................... v LIST OF FIGURES ......................................................................................................................
    [Show full text]
  • Calculus of Variations
    MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 16.323 Lecture 5 Calculus of Variations • Calculus of Variations • Most books cover this material well, but Kirk Chapter 4 does a particularly nice job. • See here for online reference. x(t) x*+ αδx(1) x*- αδx(1) x* αδx(1) −αδx(1) t t0 tf Figure by MIT OpenCourseWare. Spr 2008 16.323 5–1 Calculus of Variations • Goal: Develop alternative approach to solve general optimization problems for continuous systems – variational calculus – Formal approach will provide new insights for constrained solutions, and a more direct path to the solution for other problems. • Main issue – General control problem, the cost is a function of functions x(t) and u(t). � tf min J = h(x(tf )) + g(x(t), u(t), t)) dt t0 subject to x˙ = f(x, u, t) x(t0), t0 given m(x(tf ), tf ) = 0 – Call J(x(t), u(t)) a functional. • Need to investigate how to find the optimal values of a functional. – For a function, we found the gradient, and set it to zero to find the stationary points, and then investigated the higher order derivatives to determine if it is a maximum or minimum. – Will investigate something similar for functionals. June 18, 2008 Spr 2008 16.323 5–2 • Maximum and Minimum of a Function – A function f(x) has a local minimum at x� if f(x) ≥ f(x �) for all admissible x in �x − x�� ≤ � – Minimum can occur at (i) stationary point, (ii) at a boundary, or (iii) a point of discontinuous derivative.
    [Show full text]
  • 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.
    [Show full text]
  • Inner Product Spaces
    CHAPTER 6 Woman teaching geometry, from a fourteenth-century edition of Euclid’s geometry book. Inner Product Spaces In making the definition of a vector space, we generalized the linear structure (addition and scalar multiplication) of R2 and R3. We ignored other important features, such as the notions of length and angle. These ideas are embedded in the concept we now investigate, inner products. Our standing assumptions are as follows: 6.1 Notation F, V F denotes R or C. V denotes a vector space over F. LEARNING OBJECTIVES FOR THIS CHAPTER Cauchy–Schwarz Inequality Gram–Schmidt Procedure linear functionals on inner product spaces calculating minimum distance to a subspace Linear Algebra Done Right, third edition, by Sheldon Axler 164 CHAPTER 6 Inner Product Spaces 6.A Inner Products and Norms Inner Products To motivate the concept of inner prod- 2 3 x1 , x 2 uct, think of vectors in R and R as x arrows with initial point at the origin. x R2 R3 H L The length of a vector in or is called the norm of x, denoted x . 2 k k Thus for x .x1; x2/ R , we have The length of this vector x is p D2 2 2 x x1 x2 . p 2 2 x1 x2 . k k D C 3 C Similarly, if x .x1; x2; x3/ R , p 2D 2 2 2 then x x1 x2 x3 . k k D C C Even though we cannot draw pictures in higher dimensions, the gener- n n alization to R is obvious: we define the norm of x .x1; : : : ; xn/ R D 2 by p 2 2 x x1 xn : k k D C C The norm is not linear on Rn.
    [Show full text]