DOCSLIB.ORG

MAT641-Inner Product Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
MAT641-Inner Product Spaces MAT641- MSC Mathematics, MNIT Jaipur Functional Analysis-Inner Product Space and orthogonality Dr. Ritu Agarwal Department of Mathematics, Malaviya National Institute of Technology Jaipur April 11, 2017 1 Inner Product Space Inner product on linear space X over field K is defined as a function h:i : X × X −! K which satisfies following axioms: For x; y; z 2 X, α 2 K, i. hx + y; zi = hx; zi + hy; zi; ii. hαx; yi = αhx; yi; iii. hx; yi = hy; xi; iv. hx; xi ≥ 0, hx; xi = 0 iff x = 0. An inner product defines norm as: kxk = phx; xi and the induced metric on X is given by d(x; y) = kx − yk = phx − y; x − yi Further, following results can be derived from these axioms: hαx + βy; zi = αhx; zi + βhy; zi hx; αyi = αhx; yi hx; αy + βzi = αhx; yi + βhx; zi α, β are scalars and bar denotes complex conjugation. Remark 1.1. Inner product is linear w.r.to first factor and conjugate linear (semilin- ear) w.r.to second factor. Together, this property is called `Sesquilinear' (means one and half-linear). Date: April 11, 2017 Email:[email protected] Web: drrituagarwal.wordpress.com MAT641-Inner Product Space and orthogonality 2 Definition 1.2 (Hilbert space). A complete inner product space is known as Hilbert Space. Example 1. Examples of Inner product spaces n n i. Euclidean Space R , hx; yi = ξ1η1 + ::: + ξnηn, x = (ξ1; :::; ξn), y = (η1; :::; ηn) 2 R . n n ii. Unitary Space C , hx; yi = ξ1η1 + ::: + ξnηn, x = (ξ1; :::; ξn), y = (η1; :::; ηn) 2 C . 2 R b iii. Function space L [a; b], hx; yi = a x(t)y(t)dt, x(t) and y(t) are complex valued functions (May be real valued also). 2 P1 2 P1 iv. l = fx = (ξ1; ξ2; :::): i=1 jξij < 1g is an IPS for hx; yi = i=1 ξiη¯i. 2 Norm and Parallelogram equality To prove triangle inequality for normed space, first we are required to prove the Schwarz inequality. Theorem 2.1 (Schwarz inequality). jhx; yij ≤ kxkkyk (2.1) where, equality holds if and only if fx; yg is linearly dependent set. Proof. If y = 0, then (2.1) holds since hx; 0i = 0. Let y 6= 0. For every scalar α, we have 0 ≤ kx − αyk2 = hx − αy; x − αyi (2.2) = hx; xi − α¯hx; yi − α[hy; xi − α¯hy; yi] hy;xi Choosing α = hy;yi so that the expression in brackets vanish. We get hy; xi jhx; yij2 0 ≤ hx; xi − hx; yi = kxk2 − (Usinghy; xi = hx; yi) (2.3) hy; yi kyk2 Rearranging the terms, we get (2.1). Equality holds if kx − αyk = 0 i.e. x = αy, fx; yg are linearly dependent. Theorem 2.2. All inner product spaces are normed spaces for the norm defined by kxk = phx; xi. Also, show that the triangle inequality for norm can be obtained from inner product as kx + yk ≤ kxk + kyk (2.4) where, equality holds if and only if fx; yg is linearly dependent set x = cy, c ≥ 0 or y = 0. MAT641-Inner Product Space and orthogonality 3 Proof. (i) From axiom (iv) for inner product space, we have kxk = phx; xi ≥ 0 and kxk = phx; xi = 0 iff x = 0. (ii) kαxk = phαx; αxi = pαα¯hx; xi = pjαj2hx; xi = jαj kxk. (iii) Triangle inequality kx + yk2 = hx + y; x + yi = hx; xi + hx; yi + hy; xi + hy; yi (2.5) = kxk2 + hx; yi + hy; xi + kyk2 Using Schwarz inequality (2.1) kx + yk2 ≤ kxk2 + 2jhx; yij + kyk2 ≤ kxk2 + 2kxk kyk + kyk2 = (kxk + kyk)2 (2.6) Taking square root on both sides, we get(2.4). Further, equality holds in this derivation if and only if hx; yi + hy; xi = 2kxk kyk i.e. hx; yi + hy; xi = 2<hx; yi = 2kxk kyk. But Schwarz inequality says that kxk kyk ≥ jhx; yij, so that <hx; yi ≥ jhx; yij. Since the real part of a complex number cannot exceed the absolute value, we must have equality, which implies linear dependence from (2.1), say y = 0 or x = cy. With the equality sign, we have <hx; yi = jhx; yij, and hence imaginary part must be zero. If x = cy, <(hx; yi) = <(hcy; yi) = <(chy; yi) ≥ 0 ) c > 0. Remark 2.3. Every Hilbert space is Banach space. However converse is not necessarily true. 2.1 Parallelogram equality A norm on an inner product space satisfies the important Parallelogram equality kx + yk2 + kx − yk2 = 2(kxk2 + kyk2) (2.7) Proof. LHS = kx + yk2 + kx − yk2 = hx + y; x + yi + hx − y; x − yi = hx; xi + hx; yi + hy; xi + hy; yi + hx; xi − hx; yi + hy; yi − hy; xi (2.8) = 2(hx; xi + hy; yi) = 2(kxk2 + kyk2) MAT641-Inner Product Space and orthogonality 4 Examples of normed spaces which are not inner product spaces: Example 2. Sequence space lp, p 6= 2 Consider x = (1; 1; 0; 0; 0; ::::) and y = (1; −1; 0; 0; 0; :::) 2 lp and calculate kxk = kyk = 21=p; kx + yk = kx − yk = 2 Parallelogram equality is not satisfied. Hence lp, p 6= 2 is a Banach space but not Hilbert space. Example 3. Space C[a,b] is not an inner product space for kxk = maxt2J jx(t)j, J = [a; b]. Consider x(t) = 1 and y(t) = (t − a)=(b − a). We have, kxk = 1, kyk = 1 and t−a t−a kx(t) + y(t)k = k1 + b−ak = 2, kx(t) − y(t)k = k1 − b−ak = 1. Thus, Parallelogram equality is not satisfied. Theorem 2.4 (Polarization identity). The inner product in term of induced norm is given by 1. For a real IPS X, 1 hx; yi = (kx + yk2 − kx − yk2) (2.9) 4 2. For a complex IPS 1 <hx; yi = (kx + yk2 − kx − yk2) (2.10) 4 and 1 =hx; yi = (kx + iyk2 − kx − iyk2) (2.11) 4 Proof is left as an exercise to the reader. 3 Continuity in IPS Theorem 3.1 (Continuity). If in an inner product space xn ! x and yn ! y, then hxn; yni ! hx; yi. Proof. Consider jhxn; yni − hx; yij = jhxn; yni − hxn; yi + hxn; yi − hx; yij = jhxn; yn − yij + jhxn − x; yij Applying Schwarz inequality jhxn; yni − hx; yij ≤ kxnk kyn − yk + kxn − xk kyk ! 0 as n ! 1 Hence inner product is a continuous function. MAT641-Inner Product Space and orthogonality 5 Definition 3.2 (Isomorphism). An isomorphism T of an inner product space (X; h:i) ~ onto an IPS (X; h:i0) over the same field K of scalars is a bijective linear operator ~ T : X ! X which preserves the values of the inner products i.e. hT x; T yi0 = hx; yi. Exercise 3.3. Let Y be a subspace of a Hilbert space H. 1. Y is complete iff Y is closed in H. 2. If Y is a finite dimensional subspace then Y is complete. 3. Every subspace of a separable space is separable. Theorem 3.4 (Minimizing Vector). Let X be an inner product space and Y 6= φ is a convex subset which is complete (in the metric induced by the inner product) subset. Then for every given x 2 X, there is an unique y 2 Y such that δ = inf kx − y0k = kx − yk: y02Y 0 Proof: (Existence of y): Given δ = infy02Y kx − y k. By definition of infimum, there exists a sequence (yn) in Y such that δn = kx − ynk δn ! δ as n ! 1 We will show that (yn) is a Cauchy sequence in Y. Let x − yn = vn. Then (yn + ym) kvn + vmk = k2x − (yn + ym)k = 2 x − 2 But yn; ym 2 Y ,(yn + ym)=2 2 Y as Y is convex. Hence kvn + vmk ≥ 2δ: (3.1) 2 2 2 2 2 Now kyn − ymk = kvn − vmk = 2kvnk + 2kvmk − kvn − vmk (by Parallelogram law). 2 2 2 2 2 2 2 ) kyn − ymk = 2kx − ynk + 2kx − ymk − kvn − vmk ≤ 2δn + 2δm − 4δ (using (3.1)). As n; m ! 1, δn ! δ and δm ! δ. Thus kyn − ymk ! 0 as n; m ! 1 i.e. (yn) is a Cauchy sequence. Since Y is complete, (yn) ! y 2 Y . Thus existence of y is guaranteed. (Uniqueness): Let y0 and y1 2 Y such that δ = kx − y0k = kx − y1k 2 2 2 . Now ky1 − y0k = k(y1 − x) − (y0 − x)k ≤ 2ky1 − xk + 2ky0 − xk − ky1 − x + y0 − xk (by Parallelogram law) 2 2 2 ) ky1 − y0k ≤ 2δ + 2δ − 4δ ! 0. But ky1 − y0k ≥ 0, we have ky1 − y0k = 0 i.e. y1 = y0. MAT641-Inner Product Space and orthogonality 6 4 Orthogonality Definition 4.1 (Orthogonality). An element x of an inner product space X is said to be orthogonal to an element y 2 X if hx; yi = 0 and we write x?y. Theorem 4.2 (Orthogonality). Let X be an inner product space and Y 6= φ is a convex complete (in the metric induced by the inner product) subset. Then for a fixed x 2 X, z = x − y is orthogonal to Y where kx − yk = δ = inf kx − y0k; y 2 Y: y02Y Proof: Let z ? Y is false. Then there exist a y0 2 Y such that hz; y0i = β 6= 0 Clearly, y0 6= 0 since otherwise hz; y0i = 0. Consider the norm kz − αy0k2 = hz − αy0; z − αy0i = hz; zi − αhz; y0i − α(hz; y0i − αhy0; y0i) hz;y0i β Choose α = hy0;y0i = ky0k2 so that ββ kz − αy0k2 = kzk2 − < kzk2 = δ2 ky0k2 We claim, that kz − αy0k2hδ2 can never happen.
Load more

Recommended publications
  • On Domain of Nörlund Matrix
    mathematics Article On Domain of Nörlund Matrix Kuddusi Kayaduman 1,* and Fevzi Ya¸sar 2 1 Faculty of Arts and Sciences, Department of Mathematics, Gaziantep University, Gaziantep 27310, Turkey 2 ¸SehitlerMah. Cambazlar Sok. No:9, Kilis 79000, Turkey; [email protected] * Correspondence: [email protected] Received: 24 October 2018; Accepted: 16 November 2018; Published: 20 November 2018 Abstract: In 1978, the domain of the Nörlund matrix on the classical sequence spaces lp and l¥ was introduced by Wang, where 1 ≤ p < ¥. Tu˘gand Ba¸sarstudied the matrix domain of Nörlund mean on the sequence spaces f 0 and f in 2016. Additionally, Tu˘gdefined and investigated a new sequence space as the domain of the Nörlund matrix on the space of bounded variation sequences in 2017. In this article, we defined new space bs(Nt) and cs(Nt) and examined the domain of the Nörlund mean on the bs and cs, which are bounded and convergent series, respectively. We also examined their inclusion relations. We defined the norms over them and investigated whether these new spaces provide conditions of Banach space. Finally, we determined their a-, b-, g-duals, and characterized their matrix transformations on this space and into this space. Keywords: nörlund mean; nörlund transforms; difference matrix; a-, b-, g-duals; matrix transformations 1. Introduction 1.1. Background In the studies on the sequence space, creating a new sequence space and research on its properties have been important. Some researchers examined the algebraic properties of the sequence space while others investigated its place among other known spaces and its duals, and characterized the matrix transformations on this space.
    [Show full text]
  • Arxiv:1911.05823V1 [Math.OA] 13 Nov 2019 Al Opc Oooia Pcsadcommutative and Spaces Topological Compact Cally Edt E Eut Ntplg,O Ipie Hi Ros Nteothe the in Study Proofs
    TOEPLITZ EXTENSIONS IN NONCOMMUTATIVE TOPOLOGY AND MATHEMATICAL PHYSICS FRANCESCA ARICI AND BRAM MESLAND Abstract. We review the theory of Toeplitz extensions and their role in op- erator K-theory, including Kasparov’s bivariant K-theory. We then discuss the recent applications of Toeplitz algebras in the study of solid state sys- tems, focusing in particular on the bulk-edge correspondence for topological insulators. 1. Introduction Noncommutative topology is rooted in the equivalence of categories between lo- cally compact topological spaces and commutative C∗-algebras. This duality allows for a transfer of ideas, constructions, and results between topology and operator al- gebras. This interplay has been fruitful for the advancement of both fields. Notable examples are the Connes–Skandalis foliation index theorem [17], the K-theory proof of the Atiyah–Singer index theorem [3, 4], and Cuntz’s proof of Bott periodicity in K-theory [18]. Each of these demonstrates how techniques from operator algebras lead to new results in topology, or simplifies their proofs. In the other direction, Connes’ development of noncommutative geometry [14] by using techniques from Riemannian geometry to study C∗-algebras, led to the discovery of cyclic homol- ogy [13], a homology theory for noncommutative algebras that generalises de Rham cohomology. Noncommutative geometry and topology techniques have found ample applica- tions in mathematical physics, ranging from Connes’ reformulation of the stan- dard model of particle physics [15], to quantum field theory [16], and to solid-state physics. The noncommutative approach to the study of complex solid-state sys- tems was initiated and developed in [5, 6], focusing on the quantum Hall effect and resulting in the computation of topological invariants via pairings between K- theory and cyclic homology.
    [Show full text]
  • 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).
    [Show full text]
  • Ideal Convergence and Completeness of a Normed Space
    mathematics Article Ideal Convergence and Completeness of a Normed Space Fernando León-Saavedra 1 , Francisco Javier Pérez-Fernández 2, María del Pilar Romero de la Rosa 3,∗ and Antonio Sala 4 1 Department of Mathematics, Faculty of Social Sciences and Communication, University of Cádiz, 11405 Jerez de la Frontera, Spain; [email protected] 2 Department of Mathematics, Faculty of Science, University of Cádiz, 1510 Puerto Real, Spain; [email protected] 3 Department of Mathematics, CASEM, University of Cádiz, 11510 Puerto Real, Spain 4 Department of Mathematics, Escuela Superior de Ingeniería, University of Cádiz, 11510 Puerto Real, Spain; [email protected] * Correspondence: [email protected] Received: 23 July 2019; Accepted: 21 September 2019; Published: 25 September 2019 Abstract: We aim to unify several results which characterize when a series is weakly unconditionally Cauchy (wuc) in terms of the completeness of a convergence space associated to the wuc series. If, additionally, the space is completed for each wuc series, then the underlying space is complete. In the process the existing proofs are simplified and some unanswered questions are solved. This research line was originated in the PhD thesis of the second author. Since then, it has been possible to characterize the completeness of a normed spaces through different convergence subspaces (which are be defined using different kinds of convergence) associated to an unconditionally Cauchy sequence. Keywords: ideal convergence; unconditionally Cauchy series; completeness ; barrelledness MSC: 40H05; 40A35 1. Introduction A sequence (xn) in a Banach space X is said to be statistically convergent to a vector L if for any # > 0 the subset fn : kxn − Lk > #g has density 0.
    [Show full text]
  • On Some Bk Spaces
    IJMMS 2003:28, 1783–1801 PII. S0161171203204324 http://ijmms.hindawi.com © Hindawi Publishing Corp. ON SOME BK SPACES BRUNO DE MALAFOSSE Received 25 April 2002 ◦ (c) We characterize the spaces sα(∆), sα(∆),andsα (∆) and we deal with some sets generalizing the well-known sets w0(λ), w∞(λ), w(λ), c0(λ), c∞(λ),andc(λ). 2000 Mathematics Subject Classification: 46A45, 40C05. 1. Notations and preliminary results. For a given infinite matrix A = (anm)n,m≥1, the operators An, for any integer n ≥ 1, are defined by ∞ An(X) = anmxm, (1.1) m=1 where X = (xn)n≥1 is the series intervening in the second member being con- vergent. So, we are led to the study of the infinite linear system An(X) = bn,n= 1,2,..., (1.2) where B=(b n)n≥1 is a one-column matrix and X the unknown, see [2, 3, 5, 6, 7, 9]. Equation (1.2) can be written in the form AX = B, where AX = (An(X))n≥1.In this paper, we will also consider A as an operator from a sequence space into another sequence space. A Banach space E of complex sequences with the norm E is a BK space if P X → P X E each projection n : n is continuous. A BK space is said to have AK B = (b ) B = ∞ b e (see [8]) if for every n n≥1, n=1 m m, that is, ∞ bmem → 0 (n →∞). (1.3) m=N+1 E We shall write s, c, c0,andl∞ for the sets of all complex, convergent sequences, sequences convergent to zero, and bounded sequences, respectively.
    [Show full text]
  • Sufficient Generalities About Topological Vector Spaces
    (November 28, 2016) Topological vector spaces Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/~garrett/m/fun/notes 2016-17/tvss.pdf] 1. Banach spaces Ck[a; b] 2. Non-Banach limit C1[a; b] of Banach spaces Ck[a; b] 3. Sufficient notion of topological vector space 4. Unique vectorspace topology on Cn 5. Non-Fr´echet colimit C1 of Cn, quasi-completeness 6. Seminorms and locally convex topologies 7. Quasi-completeness theorem 1. Banach spaces Ck[a; b] We give the vector space Ck[a; b] of k-times continuously differentiable functions on an interval [a; b] a metric which makes it complete. Mere pointwise limits of continuous functions easily fail to be continuous. First recall the standard [1.1] Claim: The set Co(K) of complex-valued continuous functions on a compact set K is complete with o the metric jf − gjCo , with the C -norm jfjCo = supx2K jf(x)j. Proof: This is a typical three-epsilon argument. To show that a Cauchy sequence ffig of continuous functions has a pointwise limit which is a continuous function, first argue that fi has a pointwise limit at every x 2 K. Given " > 0, choose N large enough such that jfi − fjj < " for all i; j ≥ N. Then jfi(x) − fj(x)j < " for any x in K. Thus, the sequence of values fi(x) is a Cauchy sequence of complex numbers, so has a limit 0 0 f(x). Further, given " > 0 choose j ≥ N sufficiently large such that jfj(x) − f(x)j < " .
    [Show full text]
  • Some New Pascal Sequence Spaces
    Fundamental Journal of Mathematics and Applications, 1 (1) (2018) 61-68 Fundamental Journal of Mathematics and Applications Journal Homepage: www.dergipark.gov.tr/fujma Some new Pascal sequence spaces Harun Polata* aDepartment of Mathematics, Faculty of Science and Arts, Mus¸Alparslan University, Mus¸, Turkey *Corresponding author E-mail: [email protected] Article Info Abstract Keywords: a-, b- and g-duals and ba- The main purpose of the present paper is to study of some new Pascal sequence spaces sis of sequence, Matrix mappings, Pas- p¥, pc and p0. New Pascal sequence spaces p¥, pc and p0 are found as BK-spaces and cal sequence spaces it is proved that the spaces p¥, pc and p0 are linearly isomorphic to the spaces l¥, c and 2010 AMS: 46A35, 46A45, 46B45 c0 respectively. Afterward, a-, b- and g-duals of these spaces pc and p0 are computed Received: 27 March 2018 and their bases are consructed. Finally, matrix the classes (pc : lp) and (pc : c) have been Accepted: 25 June 2018 characterized. Available online: 30 June 2018 1. Preliminaries, background and notation By w, we shall denote the space all real or complex valued sequences. Any vector subspace of w is called a sequence space. We shall write l¥, c, and c0 for the spaces of all bounded, convergent and null sequence are given by l¥ = x = (xk) 2 w : sup jxkj < ¥ , k!¥ c = x = (xk) 2 w : lim xk exists and c0 = x = (xk) 2 w : lim xk = 0 . Also by bs, cs, l1 and lp we denote the spaces of all bounded, k!¥ k!¥ convergent, absolutely convergent and p-absolutely convergent series, respectively.
    [Show full text]
  • On Generalized Means and Some Related Sequence Spaces M
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Computers and Mathematics with Applications 61 (2011) 988–999 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa On generalized means and some related sequence spaces M. Mursaleen ∗, Abdullah K. Noman Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India article info a b s t r a c t Article history: In the present paper, we introduce the notion of generalized means and define some related Received 28 May 2010 sequence spaces which include the most known sequence spaces as special cases. Further, Accepted 21 December 2010 we study some topological properties of the spaces of generalized means and construct their bases. Finally, we determine the β-duals of these spaces and characterize some related Keywords: matrix classes. Sequence space ' 2010 Elsevier Ltd. All rights reserved. BK space Schauder basis β-dual Matrix transformation 1. Background, notations and preliminaries 2 D D 1 By w, we denote the space of all complex sequences. If x w, then we simply write x .xk/ instead of x .xk/kD0. Also, we write φ for the set of all finite sequences that terminate in zeros. Further, we shall use the conventions that .k/ e D .1; 1; 1; : : :/ and e is the sequence whose only non-zero term is 1 in the kth place for each k 2 N, where N D f0; 1; 2;:::g. Any vector subspace of w is called a sequence space.
    [Show full text]
  • Topologies of Some New Sequence Spaces, Their Duals, and the Graphical Representations of Neighborhoods ∗ Eberhard Malkowsky A, , Vesna Velickoviˇ C´ B
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 158 (2011) 1369–1380 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Topologies of some new sequence spaces, their duals, and the graphical representations of neighborhoods ∗ Eberhard Malkowsky a, , Vesna Velickoviˇ c´ b a Department of Mathematics, Faculty of Sciences, Fatih University, 34500 Büyükçekmece, Istanbul, Turkey b Department of Mathematics, Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000 Niš, Serbia article info abstract MSC: Many interesting topologies arise in the theory of FK spaces. Here we present some new 40H05 results on the topologies of certain FK spaces and the determinations of their dual spaces. 54E35 We also apply our software package for visualization and animations in mathematics and its extensions to the representation of neighborhoods in various topologies. Keywords: © 2011 Elsevier B.V. All rights reserved. Topologies and neighborhoods FK spaces Dual spaces Visualization Computer graphics 1. Introduction This paper deals with two subjects in mathematics and computer graphics, namely the study of certain linear topological spaces and the graphical representation of neighborhoods in the considered topologies. Many important topologies arise in the theory of sequence spaces, in particular, in FK and BK spaces and their dual spaces. Here we present some new results on the topology of certain FK spaces and their dual spaces. Our results have an interesting and important application in crystallography, in particular in the growth of crystals. According to Wulff’s principle [1], the shape of a crystal is uniquely determined by its surface energy function.
    [Show full text]
  • Functional Analysis
    Functional Analysis Lecture Notes of Winter Semester 2017/18 These lecture notes are based on my course from winter semester 2017/18. I kept the results discussed in the lectures (except for minor corrections and improvements) and most of their numbering. Typi- cally, the proofs and calculations in the notes are a bit shorter than those given in class. The drawings and many additional oral remarks from the lectures are omitted here. On the other hand, the notes con- tain a couple of proofs (mostly for peripheral statements) and very few results not presented during the course. With `Analysis 1{4' I refer to the class notes of my lectures from 2015{17 which can be found on my webpage. Occasionally, I use concepts, notation and standard results of these courses without further notice. I am happy to thank Bernhard Konrad, J¨orgB¨auerle and Johannes Eilinghoff for their careful proof reading of my lecture notes from 2009 and 2011. Karlsruhe, May 25, 2020 Roland Schnaubelt Contents Chapter 1. Banach spaces2 1.1. Basic properties of Banach and metric spaces2 1.2. More examples of Banach spaces 20 1.3. Compactness and separability 28 Chapter 2. Continuous linear operators 38 2.1. Basic properties and examples of linear operators 38 2.2. Standard constructions 47 2.3. The interpolation theorem of Riesz and Thorin 54 Chapter 3. Hilbert spaces 59 3.1. Basic properties and orthogonality 59 3.2. Orthonormal bases 64 Chapter 4. Two main theorems on bounded linear operators 69 4.1. The principle of uniform boundedness and strong convergence 69 4.2.
    [Show full text]
  • Lecture 4 Inner Product Spaces
    Lecture 4 Inner product spaces Of course, you are familiar with the idea of inner product spaces – at least finite-dimensional ones. Let X be an abstract vector space with an inner product, denoted as “ , ”, a mapping from X X h i × to R (or perhaps C). The inner product satisfies the following conditions, 1. x + y, z = x, z + y, z , x,y,z X , h i h i h i ∀ ∈ 2. αx,y = α x,y , x,y X, α R, h i h i ∀ ∈ ∈ 3. x,y = y,x , x,y X, h i h i ∀ ∈ 4. x,x 0, x X, x,x = 0 if and only if x = 0. h i≥ ∀ ∈ h i We then say that (X, , ) is an inner product space. h i In the case that the field of scalars is C, then Property 3 above becomes 3. x,y = y,x , x,y X, h i h i ∀ ∈ where the bar indicates complex conjugation. Note that this implies that, from Property 2, x,αy = α x,y , x,y X, α C. h i h i ∀ ∈ ∈ Note: For anyone who has taken courses in Mathematical Physics, the above properties may be slightly different than what you have seen, as regards the complex conjugations. In Physics, the usual convention is to complex conjugate the first entry, i.e. αx,y = α x,y . h i h i The inner product defines a norm as follows, x,x = x 2 or x = x,x . (1) h i k k k k h i p (Note: An inner product always generates a norm.
    [Show full text]
  • Arxiv:1403.1720V1 [Math.FA] 7 Mar 2014 (1.2) Matrix ( = ∆ Let Scle H Arxdmi of Domain 49-176)
    THE SEQUENCE SPACE bv AND SOME APPLICATIONS MURAT KIRIŞCI Abstract. In this work, we give well-known results related to some properties, dual spaces and matrix trans- formations of the sequence space bv and introduce the matrix domain of space bv with arbitrary triangle matrix A. Afterward, we choose the matrix A as Cesàro mean of order one, generalized weighted mean and Riesz mean and compute α−,β−, γ−duals of these spaces. And also, we characterize the matrix classes of the new spaces. 1. Introduction N The set of all sequences denotes with ω := C := {x = (xk): x : N → C, k → xk := x(k)}, where C denotes the complex field and N = {0, 1, 2,...}. Each linear subspace of ω (with the induced addition and scalar multiplication) is called a sequence space. We will write φ, ℓ∞,c and c0 for the sets of all finite, bounded, convergent and null sequences, respectively. It obviously that these sets are subsets of ω. (n) A sequence, whose k − th term is xk, is denoted by x or (xk). By e and e , (n =0, 1, 2, ...), we denote the (n) (n) sequences such that ek =1 for k =0, 1, 2, ..., and en =1 and ek =0 for k 6= n. A coordinate space (or K−space) is a vector space of numerical sequences, where addition and scalar multi- plication are defined pointwise. That is, a sequence space X with a linear topology is called a K-space provided each of the maps pi : X → C defined by pi(x) = xi is continuous for all i ∈ N.
    [Show full text]