ON GENERALIZED n-inner PRODUCT SPACES
|
|
- Lewis Pitts
- 5 years ago
- Views:
Transcription
1 Novi Sad J Math Vol 41, No 2, 2011, ON GENERALIZED n-inner PRODUCT SPACES Renu Chugh 1, Sushma Lather 2 Abstract The primary purpose of this paper is to derive a generalized (n k) inner product with n 2, from the generalized n-inner product, which is a generalization of the definition of Misiak [3] of the n-inner product for each k {1, 2, n 1} and also provide results related to the n-normed product induced by generalized n-inner product AMS Mathematics Subject Classification (2010): Primary 46C05, 46C99; Secondary 26D15, 26D10 Key words and phrases: n-norm linear space, n-inner product space, Cauchy-Schwarz inequality, Polarization Identity, Parallelogram law, generalized n-inner product space 1 Introduction Misiak [3] has introduced an n-norm and n-inner product by the following definitions Definition 11 Let n N (natural numbers) and X be a real linear space of dimension greater than or equal to n A real valued function,, on X X X n satisfying the following four properties: (i) x 1, x 2,, x n 0 if any only if x 1, x 2,, x n are linearly dependent, (ii) x 1, x 2,, x n is invariant under any permutation, (iii) x 1, x 2,, ax n a x 1, x 2,, x n, for any a R (real), (iv) x 1, x 2,, x n 1, y + z x 1, x 2,, x n 1, y + x 1, x 2,, x n 1, z is called an n-norm on X and the pair (X,,, ) is called n-normed linear space Definition 12 Assume that n is a positive integer and X is a real vector space such that dim X n and (,,, ) is a real function defined on X n+1 such n 1 that: (i) (x 1, x 1 x 2,, x n ) 0, for any x 1, x 2,, x n X and (x 1, x 1 x 2,, x n ) 0 if and only if x 1, x 2,, x n are linearly dependent vectors; (ii) (a, b x 1,, x n 1 ) (φ(a), φ(b) π(x 1 ),, π(x n 1 )), for any a, b, x 1, x 2,, x n 1 X and for any bijections π : {x 1, x 2,, x n 1 } {x 1, x 2,, x n 1 } and φ : {a, b} {a, b}; 1 Department of Mathematics, MD University, Rohtak, India 2 Department of Mathematics, MD University, Rohtak, India, lathersushma@yahoocom
2 74 Renu Chugh, Sushma Lather (iii) If n > 1, then (x 1, x 1 x 2,, x n ) (x 2, x 2 x 1, x 3,, x n ), for any x 1, x 2,, x n X; (iv) (αa, b x 1,, x n 1 ) α(a, b x 1,, x n 1 ), for any a, b, x 1,, x n 1 X and any scalar α R; (v) (a + a 1, b x 1,, x n 1 ) (a, b x 1,, x n 1 ) + (a 1, b x 1,, x n 1 ), for any a, b, a 1, x 1,, x n 1 X Then (,,, ) is called n-inner product and (X, (,,, ) n-prehilbert n 1 n 1 space If n 1, then Definition 12 reduces to the ordinary inner product This n-inner product induces an n-norm [3] by x 1,, x n (x 1, x 1 x 2,, x n ) Trencevski and Malceski [4] gave the definition of generalized n-inner product and the Cauchy-Schwarz inequality in this space as Definition 13 Assume that n is a positive integer, X is a real vector space such that dim X n and,,,, is a real function on X 2n such that (I 1 ) a 1,, a n a 1,, a n > 0 if a 1,, a n are linearly independent vectors, (I 2 ) a 1,, a n b 1,, b n b 1,, b n a 1,, a n for any a 1,, a n, b 1,, b n X (I 3 ) λa 1,, a n b 1,, b n λ a 1,, a n b 1,, b n for any scalar λ R and any a 1,, a n, b 1,, b n X, (I 4 ) a 1,, a n b 1,, b n a σ(1),, a σ(n) b 1,, b n for any odd permutation σ in the set {1,, n} and any a 1,, a n, b 1,, b n X, (I 5 ) a 1 + c, a 2,, a n b 1,, b n a 1, a 2,, a n b 1,, b n + c, a 2,, a n b 1,, b n for any a 1,, a n, b 1,, b n, c X, (I 6 ) If a 1, b 1,, b i 1, b i+1,,, b n b 1,, b n 0 for each i {1, 2,, n}, then a 1,, a n b 1,, b n 0 for arbitrary vectors a 1,, a n Then the function,,,, is called generalized n-inner product and the pair (X,,,,, ) is called generalized n-prehilbert space The generalized n-inner product on X induces an n-norm [3] by x 1,, x n x 1,, x n x 1,, x n And Cauchy-Schwarz inequality in generalized n-inner product on X is given as a 1,, a n b 1,, b n 2 a 1,, a n a 1,, a n b 1,, b n b 1,, b n In [1] we obtain the following identities:
3 On generalized n-inner product spaces 75 Polarization identity in generalized n-inner product space as 4 x, x 2,, x n y, x 2,, x n x + y, x 2,, x n 2 x y, x 2,, x n 2 And parallelogram law in generalized n-inner product space as x + y, x 2,, x n 2 + x y, x 2,, x n 2 2 x, x 2,, x n y, x 2,, x n 2 The classical known example [4] of generalized n-inner product space is Example 14 Let X be a space with inner product Then a 1 b 1 a 1 b 2 a 1 b n a 2 b 1 a 2 b 2 a 2 b n a 1,, a n b 1,, b n a n b 1 a n b 2 a n b n defines a generalized n-inner product on X Misiak [3] generalized the definition of 2-inner product given by Gahler [4] in n-inner product Recently, Trencevski and Malceski [4] introduced the concept of generalized n-inner product as the generalization of n-inner product and obtained some related results In [1], we discussed the weak and strong convergence, and proved some identities in this space In this paper, we present a simple method to derive a generalized (n k) inner product with n 2, from the generalized n-inner product for each k {1, 2, n 1} and also provide results related to n-norm induced by generalized n-inner product The notion of orthogonality in a generalized n-inner product space can be developed by using a derived generalized inner product or inner product, just as in [1, 2, 4] 2 Main results To avoid confusion, we shall sometimes denote a generalized n-inner product by,,,,,, n and an n-norm by,,, n Theorem 21 Let (X,,,,,,, n ) be generalized n-inner product space with finite dimension d n 2 Take a linearly independent set {a 1, a 2,, a d } and define the following function,,,,, n 1 on X 2(n 1) by (21) x 1, x 2,, x n 1 y 1, y 2,, y n 1 x 1, x 2,, x n 1, a i y 1, y 2,, y n 1, a i such that this function satisfies (I 6 ), then the function,,,,,, n 1 is a generalized (n 1)-inner product on X
4 76 Renu Chugh, Sushma Lather Proof We will verify that,,,,,, n 1 satisfies the following six properties of a generalized (n 1)-inner product (i) To verify this property, suppose that x 1, x 2,, x n 1 are linearly dependent Then x 1, x 2,, x n 1, a i x 1, x 2,, x n 1, a i 0, for every i {1, 2,, d} and hence x 1, x 2,, x n 1 x 1, x 2,, x n 1 0 Conversely, suppose that then x 1, x 2,, x n 1 x 1, x 2,, x n 1 0, n x 1, x 2,, x n 1, a i x 1, x 2,, x n 1, a i 0 so x 1, x 2,, x n 1, a i x 1, x 2,, x n 1, a i 0 for each i {1, 2,, d} Hence by (I 1 ) x 1, x 2,, x n 1, a i are linearly dependent for each i {1, 2,, d} By elementary linear algebra, this can only happen if x 1, x 2,, x n 1 are linearly dependent (ii) By using (I 2 ), we have x 1, x 2,, x n 1 y 1, y 2,, y n 1 n 1 x 1, x 2,, x n 1, a i y 1, y 2,, y n 1, a i y 1, y 2,, y n 1, a i x 1, x 2,, x n 1, a i y 1, y 2,, y n 1 x 1, x 2,, x n 1 n 1 (iii) λx 1, x 2,, x n 1 y 1, y 2,, y n 1 n 1 λx 1, x 2,, x n 1, a i y 1, y 2,, y n 1, a i λ x 1, x 2,, x n 1, a i y 1, y 2,, y n 1, a i λ x 1, x 2,, x n 1 y 1, y 2,, y n 1 n 1 For any scalar λ R, using (I 3 )
5 On generalized n-inner product spaces 77 (iv) x 1, x 2,, x n 1 y 1, y 2,, y n 1 n 1 x 1, x 2,, x n 1, a i y 1, y 2,, y n 1, a i x σ(1), x σ(2),, x σ(n 1), a i y 1, y 2,, y n 1, a i x σ(1), x σ(2),, x σ(n 1) y 1, y 2,, y n 1 for any odd permutation σ in the set {1,, n} and using (I 4 ) (v) x 1 + z, x 2,, x n 1 y 1, y 2,, y n 1 n 1 x 1 + z, x 2,, x n 1, a i y 1, y 2,, y n 1, a i x 1, x 2,, x n 1, a i y 1, y 2,, y n 1, a i + z, x 2,, x n 1, a i y 1, y 2,, y n 1, a i x 1, x 2,, x n 1 y 1, y 2,, y n 1 n 1 + z, x 2,, x n 1 y 1, y 2,, y n 1 n 1 (vi) If x 1, y 1,, y j 1, y j+1,, y n 1 y 1, y 2,, y n 1 n 1 {1, 2,, n 1} 0 for each j x 1, y 1,, y j 1, y j+1,, y n 1, a i y 1, y 2,, y n 1, a i 0, hence by the orthonormal basis {a 1, a 2,, a d } and assumption of the theorem, we have the required condition that x 1, x 2,, x n 1 y 1, y 2,, y n 1 n 1 0 for arbitrary vectors x 2,, x n 1 So,,,,,,, n 1 is a generalized (n 1)-inner product on X Corollary 22 Every generalized n-inner product space is generalized (n k)- inner product space for all k 1, 2,, n 1, by induction with generalized (n k)-inner product x 1, x 2,, x n k y 1, y 2,, y n x 1, x 2,, x n k, a i1, a i2,, a ik y 1, y 2,, y n k, a i1, a i2,, a ik n i 1,i 2,,i k {1,2,,d} such that this function satisfies (I 6 ), this condition is necessary for k 1, 2,, n 2, but for k n 1 it is trivially satisfied In particular,
6 78 Renu Chugh, Sushma Lather every generalized n-inner product space induces an inner product space ie x, y x, a i1, a i2,, a in 1 y, a i1, a i2,, a in 1 n i 1,i 2,,i n 1 {1,2,,d} Corollary 23 Let,,, n be the induced n-norm from a generalized n-inner product on X Then the following function ( ) 1 x 1, x 2,, x n 1 n 1 x 1, x 2,, x n 1, a i 2 2 is an (n 1)-norm that corresponds to,,,,,, n 1 on X, and by induction we have x 1, x 2,, x n k n k ( In particular, x i 1,i 2,,i k 1,2,,d ( x 1, x 2,, x n k, a i1, a i2,, a ik 2 ) 1 2 i 1,i 2,,i k 1 1,2,,d x, a i1, a i2,, a in 1 2 ) 1 2 defines a norm that corresponds to the derived generalized inner product or inner product, on X 24 Related results on n-normed space induced from generalized n-inner product space Suppose now that (X,,,, n ) is an n-normed space and {a 1, a 2,, a d } is a linearly independent orthonormal set in X Then we can show that ( ) 1 x 1, x 2,, x n 1 n 1 x 1, x 2,, x n 1, a i 2 2 defines an (n 1)-norm on X verified as: In particular, the triangle inequality can be x + y, x 2,, x n 1 2 n 1 x + y, x 2,, x n 1 x + y, x 2,, x n 1 x + y, x 2,, x n 1, a i x + y, x 2,, x n 1, a i x + y, x 2,, x n 1, a i 2 ( x, x 2,, x n 1, a i + y, x 2,, x n 1, a i ) 2
7 On generalized n-inner product spaces 79 Thus x + y, x 2,, x n 1 n 1 ( ) 1 ( x, x 2,, x n 1, a i + y, x 2,, x n 1, a i ) 2 2 ( ) 1 ( x, x 2,, x n 1, a i 2 2 ) 1 + y, x 2,, x n 1, a i 2 2 x, x 2,, x n 1 n 1 + y, x 2,, x n 1 n 1 This inequality shows the triangle inequality in (n 1)-norm Theorem 25 If the n-norm induced by generalized n-inner product satisfies the parallelogram law x + y, x 2,, x n 2 + x y, x 2,, x n 2 2 x, x 2,, x n y, x 2,, x n 2, then the (n 1)-norm induced by generalized n-inner product given by (4) satisfies x + y, x 2,, x n x y, x 2,, x n 1 2 In particular, the derived norm satisfies 2 x, x 2,, x n y, x 2,, x n 1 2 x + y 2 + x y 2 2 x y 2 Proof Polarization Identity in a generalized n-inner product space is x, x 2,, x n y, x 2,, x n 1 4 ( x + y, x 2,, x n 2 x y, x 2,, x n 2 ) and a generalized (n 1)-inner product is derived from it with respect to {a 1, a 2,, a d } One will then realize that the derived (n 1)-norm is the induced (n 1)-norm from the derived generalized (n 1)-inner product, and hence the parallelogram law follows 3 Examples Example 31 Let X R n be equipped with the standard generalized n-inner product space x 1, y 1 x 1, y 2 x 1, y n x 2, y 1 x 2, y 2 x 2, y n x 1, x 2,, x n y 1, y 2,, y n x n, y 1 x n, y 2 x n, y n
8 80 Renu Chugh, Sushma Lather where x, y is the usual inner product on R n Then the derived generalized (n k) inner product with respect to an orthonormal basis {b 1, b 2,, b n } coincides with the standard generalized (n k)-inner product on R n, that is, x 1, x 2,, x n k y 1, y 2,, y n k x 1, y 1 x 1, y 2 x 1, y n k x 2, y 1 x 2, y 2 x 2, y n k x n k, y 1 x n k, y 2 x n k, y n k In particular, the derived generalized inner product (inner product) x, y with respect to {b 1, b 2,, b n }, which is given by is the usual inner product x, y x, b 2, b 3,, b n y, b 2, b 3,, b n + x, b 1, b 3,, b n y, b 1, b 3,, b n + + x, b 1, b 2,, b n 1 y, b 1, b 2,, b n 1 Example 32 Let X R d be equipped with the standard n-inner product as in (5), with x, y being the usual inner product on R d Then one may particularly observe that the derived inner product with respect to an orthonormal basis {b 1, b 2,, b d } is given by x, y x, b i2, b i3,, b in y, b i2, b i3,, b in {i 1,i 2,,i n } {1,2,,d} ( ) d 1 x, y, n 1 ( ) d 1 (d 1)! where n 1 (d n)!(n 1)! This derived inner product is better than the previous one in the sense that it is only a multiple of the usual inner product This example may also be extended to any finite d-dimensional inner product space X References [1] Chugh, R, Sushma, Some results in generalized n-inner product spaces International Mathematical Forum, 4 (2009), [2] Risteski, Ice B, Trencevski, KG, Principal values and principal subspaces of two subspaces of vector spaces with inner product Beitrage zur Alegbra and Geometric Contribution to Algebra and Geometry, Vol 42 No 1 (2001), [3] Misiak, A, n-inner product spaces Math Nachr 140 (1989), [4] Trencevski, K, Malceski, R, On a generalized n-inner product and the corresponding Cauchy-Schwarz inequality J Inequal Pure and Appl Math, 7(2) Art 53, 2006 Received by the editors April 28, 2009
Some Results in Generalized n-inner Product Spaces
International Mathematical Forum, 4, 2009, no. 21, 1013-1020 Some Results in Generalized n-inner Product Spaces Renu Chugh and Sushma 1 Department of Mathematics M.D. University, Rohtak - 124001, India
More informationThe Generalization of Apollonious Identity to Linear n-normed Spaces
Int. J. Contemp. Math. Sciences, Vol. 5, 010, no. 4, 1187-119 The Generalization of Apollonious Identity to Linear n-normed Spaces Mehmet Açıkgöz University of Gaziantep Faculty of Science and Arts Department
More informationSome Properties in Generalized n-inner Product Spaces
Int. Journal of Math. Analysis, Vol. 4, 2010, no. 45, 2229-2234 Some Properties in Generalized n-inner Product Spaces B. Surender Reddy Department of Mathematics, PGCS, Saifabad Osmania University, Hyderabad-500004,
More informationINNER PRODUCTS ON n-inner PRODUCT SPACES
SOOCHOW JOURNAL OF MATHEMATICS Volume 28, No. 4, pp. 389-398, October 2002 INNER PRODUCTS ON n-inner PRODUCT SPACES BY HENDRA GUNAWAN Abstract. In this note, we show that in any n-inner product space with
More informationRiesz Representation Theorem on Generalized n-inner Product Spaces
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 18, 873-882 HIKARI Ltd, www.m-hikari.com Riesz Representation Theorem on Generalized n-inner Product Spaces Pudji Astuti Faculty of Mathematics and Natural
More informationy 2 . = x 1y 1 + x 2 y x + + x n y n 2 7 = 1(2) + 3(7) 5(4) = 3. x x = x x x2 n.
6.. Length, Angle, and Orthogonality In this section, we discuss the defintion of length and angle for vectors and define what it means for two vectors to be orthogonal. Then, we see that linear systems
More informationSOME RESULTS ON 2-INNER PRODUCT SPACES
Novi Sad J. Math. Vol. 37, No. 2, 2007, 35-40 SOME RESULTS ON 2-INNER PRODUCT SPACES H. Mazaheri 1, R. Kazemi 1 Abstract. We onsider Riesz Theorem in the 2-inner product spaces and give some results in
More informationON b-orthogonality IN 2-NORMED SPACES
ON b-orthogonality IN 2-NORMED SPACES S.M. GOZALI 1 AND H. GUNAWAN 2 Abstract. In this note we discuss the concept of b-orthogonality in 2-normed spaces. We observe that this definition of orthogonality
More informationy 1 y 2 . = x 1y 1 + x 2 y x + + x n y n y n 2 7 = 1(2) + 3(7) 5(4) = 4.
. Length, Angle, and Orthogonality In this section, we discuss the defintion of length and angle for vectors. We also define what it means for two vectors to be orthogonal. Then we see that linear systems
More informationProjection Theorem 1
Projection Theorem 1 Cauchy-Schwarz Inequality Lemma. (Cauchy-Schwarz Inequality) For all x, y in an inner product space, [ xy, ] x y. Equality holds if and only if x y or y θ. Proof. If y θ, the inequality
More informationON ANGLES BETWEEN SUBSPACES OF INNER PRODUCT SPACES
ON ANGLES BETWEEN SUBSPACES OF INNER PRODUCT SPACES HENDRA GUNAWAN AND OKI NESWAN Abstract. We discuss the notion of angles between two subspaces of an inner product space, as introduced by Risteski and
More informationH. Gunawan, E. Kikianty, Mashadi, S. Gemawati, and I. Sihwaningrum
Research Report 8 December 2006 ORTHOGONALITY IN n-normed SPACES H. Gunawan, E. Kikianty, Mashadi, S. Gemawati, and I. Sihwaningrum Abstract. We introduce several notions of orthogonality in n-normed spaces.
More informationVector spaces. DS-GA 1013 / MATH-GA 2824 Optimization-based Data Analysis.
Vector spaces DS-GA 1013 / MATH-GA 2824 Optimization-based Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_fall17/index.html Carlos Fernandez-Granda Vector space Consists of: A set V A scalar
More informationAn Extension in the Domain of an n-norm Defined on the Space of p-summable Sequences
Gen. Math. Notes, Vol. 33, No., March 206, pp.-8 ISSN 229-784; Copyright c ICSRS Publication, 206 www.i-csrs.org Available free online at http://www.geman.in An Extension in the Domain of an n-norm Defined
More informationProf. M. Saha Professor of Mathematics The University of Burdwan West Bengal, India
CHAPTER 9 BY Prof. M. Saha Professor of Mathematics The University of Burdwan West Bengal, India E-mail : mantusaha.bu@gmail.com Introduction and Objectives In the preceding chapters, we discussed normed
More informationLinear Algebra. Paul Yiu. Department of Mathematics Florida Atlantic University. Fall A: Inner products
Linear Algebra Paul Yiu Department of Mathematics Florida Atlantic University Fall 2011 6A: Inner products In this chapter, the field F = R or C. We regard F equipped with a conjugation χ : F F. If F =
More informationDS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationElementary linear algebra
Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The
More informationBOUNDED LINEAR FUNCTIONALS ON THE n-normed SPACE OF p-summable SEQUENCES
BOUNDED LINEAR FUNCTIONALS ON THE n-normed SPACE OF p-summable SEQUENCES HARMANUS BATKUNDE, HENDRA GUNAWAN*, AND YOSAFAT E.P. PANGALELA Abstract. Let (X,,, ) be a real n-normed space, as introduced by
More informationLecture Notes 1: Vector spaces
Optimization-based data analysis Fall 2017 Lecture Notes 1: Vector spaces In this chapter we review certain basic concepts of linear algebra, highlighting their application to signal processing. 1 Vector
More information5 Compact linear operators
5 Compact linear operators One of the most important results of Linear Algebra is that for every selfadjoint linear map A on a finite-dimensional space, there exists a basis consisting of eigenvectors.
More informationThe following definition is fundamental.
1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic
More informationORTHOGONALITY AND LEAST-SQUARES [CHAP. 6]
ORTHOGONALITY AND LEAST-SQUARES [CHAP. 6] Inner products and Norms Inner product or dot product of 2 vectors u and v in R n : u.v = u 1 v 1 + u 2 v 2 + + u n v n Calculate u.v when u = 1 2 2 0 v = 1 0
More informationApplied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationPart 1a: Inner product, Orthogonality, Vector/Matrix norm
Part 1a: Inner product, Orthogonality, Vector/Matrix norm September 19, 2018 Numerical Linear Algebra Part 1a September 19, 2018 1 / 16 1. Inner product on a linear space V over the number field F A map,
More informationMath Linear Algebra
Math 220 - Linear Algebra (Summer 208) Solutions to Homework #7 Exercise 6..20 (a) TRUE. u v v u = 0 is equivalent to u v = v u. The latter identity is true due to the commutative property of the inner
More informationMATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.
MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v
More informationCHAPTER II HILBERT SPACES
CHAPTER II HILBERT SPACES 2.1 Geometry of Hilbert Spaces Definition 2.1.1. Let X be a complex linear space. An inner product on X is a function, : X X C which satisfies the following axioms : 1. y, x =
More informationLinear Algebra I. Ronald van Luijk, 2015
Linear Algebra I Ronald van Luijk, 2015 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents Dependencies among sections 3 Chapter 1. Euclidean space: lines and hyperplanes 5 1.1. Definition
More informationConvergent Iterative Algorithms in the 2-inner Product Space R n
Int. J. Open Problems Compt. Math., Vol. 6, No. 4, December 2013 ISSN 1998-6262; Copyright c ICSRS Publication, 2013 www.i-csrs.org Convergent Iterative Algorithms in the 2-inner Product Space R n Iqbal
More information6 Inner Product Spaces
Lectures 16,17,18 6 Inner Product Spaces 6.1 Basic Definition Parallelogram law, the ability to measure angle between two vectors and in particular, the concept of perpendicularity make the euclidean space
More informationDefinitions and Properties of R N
Definitions and Properties of R N R N as a set As a set R n is simply the set of all ordered n-tuples (x 1,, x N ), called vectors. We usually denote the vector (x 1,, x N ), (y 1,, y N ), by x, y, or
More informationNORMS ON SPACE OF MATRICES
NORMS ON SPACE OF MATRICES. Operator Norms on Space of linear maps Let A be an n n real matrix and x 0 be a vector in R n. We would like to use the Picard iteration method to solve for the following system
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week: December 4 8 Deadline to hand in the homework: your exercise class on week January 5. Exercises with solutions ) Let H, K be Hilbert spaces, and A : H K be a linear
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More informationInner products. Theorem (basic properties): Given vectors u, v, w in an inner product space V, and a scalar k, the following properties hold:
Inner products Definition: An inner product on a real vector space V is an operation (function) that assigns to each pair of vectors ( u, v) in V a scalar u, v satisfying the following axioms: 1. u, v
More informationReview of Some Concepts from Linear Algebra: Part 2
Review of Some Concepts from Linear Algebra: Part 2 Department of Mathematics Boise State University January 16, 2019 Math 566 Linear Algebra Review: Part 2 January 16, 2019 1 / 22 Vector spaces A set
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationSome Aspects of 2-Fuzzy 2-Normed Linear Spaces
BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 32(2) (2009), 211 221 Some Aspects of 2-Fuzzy 2-Normed Linear Spaces 1 R. M. Somasundaram
More information08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms
(February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops
More informationMathematics Department Stanford University Math 61CM/DM Inner products
Mathematics Department Stanford University Math 61CM/DM Inner products Recall the definition of an inner product space; see Appendix A.8 of the textbook. Definition 1 An inner product space V is a vector
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More informationMath 290, Midterm II-key
Math 290, Midterm II-key Name (Print): (first) Signature: (last) The following rules apply: There are a total of 20 points on this 50 minutes exam. This contains 7 pages (including this cover page) and
More informationorthogonal relations between vectors and subspaces Then we study some applications in vector spaces and linear systems, including Orthonormal Basis,
5 Orthogonality Goals: We use scalar products to find the length of a vector, the angle between 2 vectors, projections, orthogonal relations between vectors and subspaces Then we study some applications
More informationChapter 6 Inner product spaces
Chapter 6 Inner product spaces 6.1 Inner products and norms Definition 1 Let V be a vector space over F. An inner product on V is a function, : V V F such that the following conditions hold. x+z,y = x,y
More informationIntroduction to Numerical Linear Algebra II
Introduction to Numerical Linear Algebra II Petros Drineas These slides were prepared by Ilse Ipsen for the 2015 Gene Golub SIAM Summer School on RandNLA 1 / 49 Overview We will cover this material in
More information2. Review of Linear Algebra
2. Review of Linear Algebra ECE 83, Spring 217 In this course we will represent signals as vectors and operators (e.g., filters, transforms, etc) as matrices. This lecture reviews basic concepts from linear
More informationPolar Form of a Complex Number (I)
Polar Form of a Complex Number (I) The polar form of a complex number z = x + iy is given by z = r (cos θ + i sin θ) where θ is the angle from the real axes to the vector z and r is the modulus of z, i.e.
More informationMath 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Exam 2 will be held on Tuesday, April 8, 7-8pm in 117 MacMillan What will be covered The exam will cover material from the lectures
More informationMAT 771 FUNCTIONAL ANALYSIS HOMEWORK 3. (1) Let V be the vector space of all bounded or unbounded sequences of complex numbers.
MAT 771 FUNCTIONAL ANALYSIS HOMEWORK 3 (1) Let V be the vector space of all bounded or unbounded sequences of complex numbers. (a) Define d : V V + {0} by d(x, y) = 1 ξ j η j 2 j 1 + ξ j η j. Show that
More informationINNER PRODUCT SPACE. Definition 1
INNER PRODUCT SPACE Definition 1 Suppose u, v and w are all vectors in vector space V and c is any scalar. An inner product space on the vectors space V is a function that associates with each pair of
More informationFourier and Wavelet Signal Processing
Ecole Polytechnique Federale de Lausanne (EPFL) Audio-Visual Communications Laboratory (LCAV) Fourier and Wavelet Signal Processing Martin Vetterli Amina Chebira, Ali Hormati Spring 2011 2/25/2011 1 Outline
More informationAlgebra II. Paulius Drungilas and Jonas Jankauskas
Algebra II Paulius Drungilas and Jonas Jankauskas Contents 1. Quadratic forms 3 What is quadratic form? 3 Change of variables. 3 Equivalence of quadratic forms. 4 Canonical form. 4 Normal form. 7 Positive
More informationProjective Schemes with Degenerate General Hyperplane Section II
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (2003), No. 1, 111-126. Projective Schemes with Degenerate General Hyperplane Section II E. Ballico N. Chiarli S. Greco
More informationSOME NEW DOUBLE SEQUENCE SPACES IN n-normed SPACES DEFINED BY A SEQUENCE OF ORLICZ FUNCTION
Journal of Mathematical Analysis ISSN: 2217-3412, URL: http://www.ilirias.com Volume 3 Issue 12012, Pages 12-20. SOME NEW DOUBLE SEQUENCE SPACES IN n-normed SPACES DEFINED BY A SEQUENCE OF ORLICZ FUNCTION
More informationVectors in Function Spaces
Jim Lambers MAT 66 Spring Semester 15-16 Lecture 18 Notes These notes correspond to Section 6.3 in the text. Vectors in Function Spaces We begin with some necessary terminology. A vector space V, also
More informationMath Linear Algebra II. 1. Inner Products and Norms
Math 342 - Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationA fixed point approach to orthogonal stability of an Additive - Cubic functional equation
Int. J. Adv. Appl. Math. and Mech. 3(4 (06 8 (ISSN: 347-59 Journal homepage: www.ijaamm.com IJAAMM International Journal of Advances in Applied Mathematics and Mechanics A fixed point approach to orthogonal
More informationData representation and classification
Representation and classification of data January 25, 2016 Outline Lecture 1: Data Representation 1 Lecture 1: Data Representation Data representation The phrase data representation usually refers to the
More informationI teach myself... Hilbert spaces
I teach myself... Hilbert spaces by F.J.Sayas, for MATH 806 November 4, 2015 This document will be growing with the semester. Every in red is for you to justify. Even if we start with the basic definition
More informationInequalities for Modules and Unitary Invariant Norms
Int. J. Contemp. Math. Sciences Vol. 7 202 no. 36 77-783 Inequalities for Modules and Unitary Invariant Norms Loredana Ciurdariu Department of Mathematics Politehnica University of Timisoara P-ta. Victoriei
More informationVector Spaces, Affine Spaces, and Metric Spaces
Vector Spaces, Affine Spaces, and Metric Spaces 2 This chapter is only meant to give a short overview of the most important concepts in linear algebra, affine spaces, and metric spaces and is not intended
More informationMATH Linear Algebra
MATH 4 - Linear Algebra One of the key driving forces for the development of linear algebra was the profound idea (going back to 7th century and the work of Pierre Fermat and Rene Descartes) that geometry
More information4 Hilbert spaces. The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan
The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan Wir müssen wissen, wir werden wissen. David Hilbert We now continue to study a special class of Banach spaces,
More informationA Brief Introduction to Functional Analysis
A Brief Introduction to Functional Analysis Sungwook Lee Department of Mathematics University of Southern Mississippi sunglee@usm.edu July 5, 2007 Definition 1. An algebra A is a vector space over C with
More informationDuke University, Department of Electrical and Computer Engineering Optimization for Scientists and Engineers c Alex Bronstein, 2014
Duke University, Department of Electrical and Computer Engineering Optimization for Scientists and Engineers c Alex Bronstein, 2014 Linear Algebra A Brief Reminder Purpose. The purpose of this document
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More informationSystems of Linear Equations
Systems of Linear Equations Math 108A: August 21, 2008 John Douglas Moore Our goal in these notes is to explain a few facts regarding linear systems of equations not included in the first few chapters
More informationA matrix problem and a geometry problem
University of Szeged, Bolyai Institute and MTA-DE "Lendület" Functional Analysis Research Group, University of Debrecen Seminar University of Primorska, Koper Theorem (L. Molnár and W. Timmermann, 2011)
More informationMIDTERM I LINEAR ALGEBRA. Friday February 16, Name PRACTICE EXAM SOLUTIONS
MIDTERM I LIEAR ALGEBRA MATH 215 Friday February 16, 2018. ame PRACTICE EXAM SOLUTIOS Please answer the all of the questions, and show your work. You must explain your answers to get credit. You will be
More informationLecture 23: 6.1 Inner Products
Lecture 23: 6.1 Inner Products Wei-Ta Chu 2008/12/17 Definition An inner product on a real vector space V is a function that associates a real number u, vwith each pair of vectors u and v in V in such
More informationMath 5520 Homework 2 Solutions
Math 552 Homework 2 Solutions March, 26. Consider the function fx) = 2x ) 3 if x, 3x ) 2 if < x 2. Determine for which k there holds f H k, 2). Find D α f for α k. Solution. We show that k = 2. The formulas
More informationMATH 532: Linear Algebra
MATH 532: Linear Algebra Chapter 5: Norms, Inner Products and Orthogonality Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Spring 2015 fasshauer@iit.edu MATH 532 1 Outline
More informationLecture 5. Ch. 5, Norms for vectors and matrices. Norms for vectors and matrices Why?
KTH ROYAL INSTITUTE OF TECHNOLOGY Norms for vectors and matrices Why? Lecture 5 Ch. 5, Norms for vectors and matrices Emil Björnson/Magnus Jansson/Mats Bengtsson April 27, 2016 Problem: Measure size of
More informationFurther Mathematical Methods (Linear Algebra) 2002
Further Mathematical Methods (Linear Algebra) 22 Solutions For Problem Sheet 3 In this Problem Sheet, we looked at some problems on real inner product spaces. In particular, we saw that many different
More informationON OSCULATING, NORMAL AND RECTIFYING BI-NULL CURVES IN R 5 2
Novi Sad J. Math. Vol. 48, No. 1, 2018, 9-20 https://doi.org/10.30755/nsjom.05268 ON OSCULATING, NORMAL AND RECTIFYING BI-NULL CURVES IN R 5 2 Kazım İlarslan 1, Makoto Sakaki 2 and Ali Uçum 34 Abstract.
More informationMath 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More informationI-CONVERGENT TRIPLE SEQUENCE SPACES OVER n-normed SPACE
Asia Pacific Journal of Mathematics, Vol. 5, No. 2 (2018), 233-242 ISSN 2357-2205 I-CONVERGENT TRIPLE SEQUENCE SPACES OVER n-normed SPACE TANWEER JALAL, ISHFAQ AHMAD MALIK Department of Mathematics, National
More informationCambridge University Press The Mathematics of Signal Processing Steven B. Damelin and Willard Miller Excerpt More information
Introduction Consider a linear system y = Φx where Φ can be taken as an m n matrix acting on Euclidean space or more generally, a linear operator on a Hilbert space. We call the vector x a signal or input,
More informationDot Products. K. Behrend. April 3, Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem.
Dot Products K. Behrend April 3, 008 Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem. Contents The dot product 3. Length of a vector........................
More informationInner Product Spaces
Inner Product Spaces Introduction Recall in the lecture on vector spaces that geometric vectors (i.e. vectors in two and three-dimensional Cartesian space have the properties of addition, subtraction,
More informationHomework 2. Solutions T =
Homework. s Let {e x, e y, e z } be an orthonormal basis in E. Consider the following ordered triples: a) {e x, e x + e y, 5e z }, b) {e y, e x, 5e z }, c) {e y, e x, e z }, d) {e y, e x, 5e z }, e) {
More informationThe Four Fundamental Subspaces
The Four Fundamental Subspaces Introduction Each m n matrix has, associated with it, four subspaces, two in R m and two in R n To understand their relationships is one of the most basic questions in linear
More informationAN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES
AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral radius of a matrix A may be obtained using the formula ρ(a) lim
More informationMath 113 Solutions: Homework 8. November 28, 2007
Math 113 Solutions: Homework 8 November 28, 27 3) Define c j = ja j, d j = 1 j b j, for j = 1, 2,, n Let c = c 1 c 2 c n and d = vectors in R n Then the Cauchy-Schwarz inequality on Euclidean n-space gives
More informationLinear Analysis Lecture 5
Linear Analysis Lecture 5 Inner Products and V Let dim V < with inner product,. Choose a basis B and let v, w V have coordinates in F n given by x 1. x n and y 1. y n, respectively. Let A F n n be the
More informationMATH ASSIGNMENT 1 - SOLUTIONS September 11, 2006
MATH 0-090 ASSIGNMENT - September, 00. Using the Trichotomy Law prove that if a and b are real numbers then one and only one of the following is possible: a < b, a b, or a > b. Since a and b are real numbers
More informationSection 7.5 Inner Product Spaces
Section 7.5 Inner Product Spaces With the dot product defined in Chapter 6, we were able to study the following properties of vectors in R n. ) Length or norm of a vector u. ( u = p u u ) 2) Distance of
More informationSYLLABUS. 1 Linear maps and matrices
Dr. K. Bellová Mathematics 2 (10-PHY-BIPMA2) SYLLABUS 1 Linear maps and matrices Operations with linear maps. Prop 1.1.1: 1) sum, scalar multiple, composition of linear maps are linear maps; 2) L(U, V
More informationLinear Algebra I for Science (NYC)
Element No. 1: To express concrete problems as linear equations. To solve systems of linear equations using matrices. Topic: MATRICES 1.1 Give the definition of a matrix, identify the elements and the
More informationIntroduction to Geometry
Introduction to Geometry it is a draft of lecture notes of H.M. Khudaverdian. Manchester, 18 May 211 Contents 1 Euclidean space 3 1.1 Vector space............................ 3 1.2 Basic example of n-dimensional
More informationA Review of Linear Algebra
A Review of Linear Algebra Mohammad Emtiyaz Khan CS,UBC A Review of Linear Algebra p.1/13 Basics Column vector x R n, Row vector x T, Matrix A R m n. Matrix Multiplication, (m n)(n k) m k, AB BA. Transpose
More informationLinear Normed Spaces (cont.) Inner Product Spaces
Linear Normed Spaces (cont.) Inner Product Spaces October 6, 017 Linear Normed Spaces (cont.) Theorem A normed space is a metric space with metric ρ(x,y) = x y Note: if x n x then x n x, and if {x n} is
More informationDiscrete Math Class 19 ( )
Discrete Math 37110 - Class 19 (2016-12-01) Instructor: László Babai Notes taken by Jacob Burroughs Partially revised by instructor Review: Tuesday, December 6, 3:30-5:20 pm; Ry-276 Final: Thursday, December
More informationMATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.
MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation
More informationON SOME RESULTS ON LINEAR ORTHOGONALITY SPACES
ASIAN JOURNAL OF MATHEMATICS AND APPLICATIONS Volume 2014, Article ID ama0155, 11 pages ISSN 2307-7743 http://scienceasia.asia ON SOME RESULTS ON LINEAR ORTHOGONALITY SPACES KANU, RICHMOND U. AND RAUF,
More informationLinear Algebra: Matrix Eigenvalue Problems
CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given
More informationFUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES. Yılmaz Yılmaz
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 33, 2009, 335 353 FUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES Yılmaz Yılmaz Abstract. Our main interest in this
More information