Abstract Vector Spaces. Abstract Vector Spaces

Size: px
Start display at page:

Download "Abstract Vector Spaces. Abstract Vector Spaces"

Transcription

1 Astract Vector Spaces The process of astractio is critical i egieerig! Physical Device Data Storage Vector Space MRI machie Optical receiver Icreasig astractio 6.1 Astract Vector Spaces Dimesio of a vector space Defiitio: A vector space is -dimesioal if it possess a set of idepedet vectors, ut every set of + 1 vectors is a depedet set Defiitio: If for every +, we ca fid idepedet vectors i, the is -dimesioal. Defiitio: A fiite set of vectors e 1,, e is a asis for a vector space if: (a) The vectors e 1,, e are idepedet, ad () every vector x ca e writte as a liear comiatio of the asis vectors, i.e., x = 1 e e, where the j if is a real vector space ad j if is a real vector space Theorem: The represetatio of x i terms of a give set of asis vectors is uique. Theorem: If is -dimesioal, the ay set of idepedet vectors forms a asis 6. 1

2 Examples of vector spaces Astract Vector Spaces Example 1: The vector spaces ad. We write 0 = (0,, 0) ad the vectors e 1 = (1, 0, 0), e = (0, 1, 0), e = (0, 0,, 0) form a asis. These vector spaces are used as astract represetatio of the row vectors i matrix algera. Example : Let f 1 (t),, f (t) e ay real fuctios. All fuctios of the form 1 f 1 (t) + + f (t), costitute a -dimesioal vector space. I particular, we see that all polyomials 1 + t + + t of degree < costitute a -dimesioal vector space Example 3 (Ifiite dimesioal spaces): (a) All real- or complex-valued fuctios x(t) defied i a t. () The space of all real- or complexvalued fuctios for which x dt exists. This space is -dimesioal. a 6.3 Metric Spaces ad Norms Metric spaces: Defiitio: A collectio of poits (ot ecessarily a vector space) is called a metric space if to each pair of elemets x ad y, there correspods a umer d(x, y) that satisfies the properties: (a) d(x, y) = d(y, x); () d(x, y) 0, d(x, y) = 0 if ad oly if x = y; (c) d(x, z) d(x, y) + d(y, z) [triagle iequality] Defiitio: Cosider a sequece of elemets {x }. We write x x or x coverges to x if for each > 0, there exists a idex N such that d(x, x ) wheever > N. Ofte, we do ot ow if the elemets coverge to a fixed elemet i the space. So, we defie a Cauchy sequece. Defiitio: A sequece {x } is a Cauchy sequece if to each > 0, there exists a idex N such that d(x m, x p ) < wheever m, p > N. If {x } is a Cauchy sequece, we will write lim d( x, x ) 0 mp, m p 6.4

3 Metric spaces: Metric Spaces ad Norms Theorem: If a sequece coverges, it is a Cauchy sequece Proof: Let x e the limit of a sequece. By the triagle iequality, we have d(x m, x p ) d(x m, x) + d(x, x p ). Sice x x, there is some N, such that d(x, x) wheever > N. Hece, if m ad p are oth larger tha N, the d(x m, x p ) NOTE: The coverse does ot ecessarily hold! Defiitio: A metric space is complete if every Cauchy sequece is a coverget sequece. Remar: If the metric space is ot complete, we may add the limits of the Cauchy sequeces to complete the space. 6.5 Metric spaces: Metric Spaces ad Norms Example of icompleteess: The space of ratioal umers with the metric d(x, y) = x y is a metric space, ut it is ot complete Cosider the sequece x1 1, x 1,, x 1 1! 1!! ( 1)! Its limit is e, which is irratioal. Hece, the Cauchy series does ot coverge to a elemet i the space. A similar coclusio ca e reached for the series x1, x,, x ( 1)( 3) which coverges to /8. Remar: By cotrast, the space of real umers is complete with this metric

4 Examples of metric spaces: Metric Spaces ad Norms Example 1: The space of real or the space of complex umers with the metric d(x, y) = x y are oth complete metric spaces. Example : The spaces ad with the metric d( x, y) 1/ 1 1 where x = ( 1,,, ) ad y = ( 1,,, ) are complete metric spaces Example 3: The space C(a, ) of all cotiuous complex-valued fuctios. With the metric d1 ( x, y) max x( t) y( t), the space is complete. This metric a t assures us that if ay Cauchy sequece will coverge uiformly. Ay uiformly coverget sequece of cotiuous fuctios is cotiuous. With the metric d ( x, y) x() t y() t dt, the space is ot complete. 1/ a 6.7 Metric Spaces ad Norms Examples of metric spaces: Example 3 (cotiued): The space C(a, ) of all cotiuous complex-valued 1/ a fuctios is ot complete with the metric d ( ) ( ( x, y ) x() t y () t dt. Examples of icompleteess: (a) Cosider the sequece (with a = 1 ad = 1) 1 1 x ( t) arcta t, 1 t 1 it coverges to: y(t) = 1, 0 < t 1; y(t) = 0, 1 t < 0, which is ot cotiuous. The itegral equals 0 for all ad hece the Cauchy sequece coverges

5 Examples of metric spaces: Metric Spaces ad Norms Example 3 (cotiued): The space C(a, ) of all cotiuous complex-valued 1/ a fuctios is ot complete with the metric d ( x, y) x() t y() t dt. Examples of icompleteess: () Cosider the sequece (with a = ad = 1) 1; t r/ s, 1 s1, 0 r s x () t 0; otherwise It coverges to a fuctio that equals 0 whe t is irratioal ad equals 1 whe t is ratioal. The stadard Riema itegral of elemetary calculus exists for all ad yields zero. Hece, the Cauchy sequece coverges to zero. However, the Riema itegral does ot exist! We must use the Leesgue itegral!! 6.9 Metric Spaces ad Norms Properties of the Leesgue itegral: (1) The space of all square-leesgue-itegrale fuctio (a, ) completes the space C(a, ) of all cotiuous complex-valued fuctios with the metric 1/ d ( x, y) x() t y() t dt. a Hece, the space C(a, ) is dese i (a, ) ad ay elemet i (a, ) ca e approximated y a Cauchy sequece of elemets i C(a, ). () The Riema itegral of all Riema-itegrale fuctios equals the Leesgue itegral. I practice, itegratio always meas Leesgue itegratio (3) Two fuctios are equal almost everywhere if d (x, y) = 0. A set of fuctios that are equal almost everywhere are treated as a sigle poit i (a, )

6 Normed vector spaces: Metric Spaces ad Norms Defiitio: A ormed vector space is a vector space with a real-valued fuctio x with the properties: (1) x 0, x = 0 if ad oly if x = 0; () x = x ; (3) x + y x + y Remars: A ormed vector space is a metric space with the atural metric d(x, y) = x y. The coverse is ot true; d(x, y) = 0 whe x = y ad d(x, y) = 1, otherwise is a metric, ut it is ot a orm (Why?) Defiitio: A ormed vector space that is complete i its atural orm is a Baach space Ier Product Spaces Defiitio: A ier product x, y o a real vector space (r) is a real-valued fuctio of a order pair of vectors x, y with the properties: (1) x, y = y, x; () x, y = x, y; (3) x 1 + x, y = x 1, y + x, y; (4) x, x 0, with equality holdig oly for x = 0 Remar: A ier product ca also e defied o complex vector spaces. The oly chage is that x, y y, x. Note that x, y = x, y Theorem (Schwarz Iequality): For ay two vectors x ad y, we have Proof: We write x, y x, x y, y 0,,,,, x y x y y y y x x y x x We ow let = r x, y/ x, y, where r is a aritrary real umer, ad we fid 0,,, r y y r x y x x which is oly possile for all r if the theorem holds

7 Ier Product Spaces Theorem (Schwarz Iequality): For ay two vectors x ad y, we have x, y x, x y, y Proof (cotiued): The quadratic o the right is a miimum whe r x, y y, y The theorem follows Remar 1: I three dimesios, this theorem ecomes a a cos Remar : We ow fid 1/ 1/ 1/ 1/ x y, x y x, x Re x, y y, y x, x y, y It follows that x x, x 1/ defies a atural orm for the ier product space. The Schwarz iequality may ow e writte x, y x y 6.13 Hilert Space: Ier Product Spaces Defiitio: A ier product space that is complete i its atural orm is a Hilert space. Defiitio: A Hilert space is separale if there are a coutale umer of elemets f 1, f,, f, such that, give f i ad > 0, there exists a N that satisfies N f f 1 Remar:,, ad (a, ) are separale Hilert spaces. The ier products for or ad (a, ) are x, y x y, x, y x( t) y( t) dt 1 a Separaility is ovious for or, ut must e proved for (a, )

8 Ier Product Spaces Hilert Space: Remar: The atural orm for these spaces is the -orm 1/ 1/ x x, x x( t) dt a 1 Other importat orms are the p-orm (particularly p = 1) 1/ p p p x, ( ) p x x x t dt p a 1 ad the -orm, x max x, x max x( t) 1/ p 6.15 Orthogoality: Ier Product Spaces Defiitio: Two vectors are orthogoal if x, y = 0. Theorem 1: Two orthogoal vectors satisfy the Pythagorea theorem, x + y = x + y Theorem : A orthogoal set of o-zero vectors is idepedet. Hece, orthogoal vectors are a asis for a -dimesioal vector space. Theorem 3. Cosider the lie geerated y y, which is all vectors y, the ay vector x may e uiquely decomposed ito the sum of a compoet x p that proportioal to y ad a compoet z that is orthogoal to y. The compoet x p is called the projectio of x o y. We have specifically xy, xy, x xp z, xp y, z x y, z, y 0 y y

9 Gram-Schmidt procedure: Orthoormal ases Defiitio: A liear maifold is a suset of a vector space that is itself a vector space. Example: Cosider the vector space geerated y idepedet vectors x 1,, x m, where m <. The, is a liear maifold ad so is, the space that cosists of all vectors that are orthogoal to. Developig a orthoormal asis allows us to separate matrix operatios ad more geeral trasformatios ito their rage ad ull spaces! 6.17 Gram-Schmidt procedure: Orthoormal ases Give idepedet vectors {e 1, e,, e }, we wish to costruct a orthoormal set { 1,,, } Step 1: We let 1 = e 1 / e 1 Step : We let g = e e, 1 1, = g / g. We fid that 1 1, 1 g, 1 e, 1 e, 1 0 g g Cotiuig iductively, we fid for all j Step j: We let g j = e j e j, 1 1 e j, e j, j j j = g j / g j, l, m = lm (l, m j )* *The fuctio lm is called the Kroeecer delta-fuctio. It equals 1 whe l = m ad is zero otherwise

10 Orthoormal ases Gram-Schmidt procedure: Example: We let e 1 = (1,, 3), e = (1, 1, ), e 3 = (,, ), 1 3 1,, (0.67, 0.535, 0.80) g 3 11 ( 1) (1, 1, ),,,, ,, (0.313, 0.835, 0.45) g 3 (,, ) 3.1 (0.67, 0.535, 0.80) ( 0.139) )( (0.313, 0.835, 0.45) 3 (1.19, 0.170, 0.509) (0.911, 0.130, 0.391) 6.19 Gram-Schmidt procedure: Example: We let e 1 = (1,, 3), e = (1, 1, ), e 3 = (,, ) I MATLAB: Orthoormal ases >> E = [1 1 ; -1 ; 3 ], F = E >> F(1:3,1)=F(1:3,1)/orm(F(1:3,1)) >> F(1:3,) = F(1:3,) - (F(1:3,)'*F(1:3,1))*F(1:3,1) >> F(1:3,) = F(1:3,)/orm(F(1:3,)) >> F(1:3,3) = F(1:3,3) - (F(1:3,3)'*F(1:3,1))*F(1:3,1) - (F(1:3,3)'*F(1:3,))*F(1:3,) >> F(1:3,3) = F(1:3,3)/orm(F(1:3,3)) Remar: The classic Gram-Schmidt procedure ca e used to produce a QR decompositio (ut is a computatioally poor way to do it) Compare i MATLAB: >> G = F *E >> [Q, R] = qr(e)

11 Gram-Schmidt procedure: Orthoormal ases Similar ideas ca e used to costruct a orthoormal asis for (a, ). I fuctioal otatio, we have g () t e () t () t e () t () t dt () t e () t () t dt, a a 1/ () t g() t g() t dt a Example: From the Weierstrass approximatio theorem, we ow that the polyomials are dese i C(a, ) ad hece i (a, ). Startig with the polyomial set, e = = = 0 1, e 1 t,, e t, o the iterval 11 t 1 ad ormalizig so that all elemets equal 1 at t = 1, we otai the Legedre polyomials 1 0() t 1, 1() t t, () t 3t 1, 6.1 Gram-Schmidt procedure: Orthoormal ases I may applicatios, we wat to use a complete orthoormal asis i (a, ) to study how a trasformatio affects a iput sigal. A complete asis is oe for which we may write x j j, where j = x, j. Calculatig j0 j j j j1 j1 x x, x x,, we otai Bessel s iequality x x, j, j11 The goal is to show that as, the iequality ecomes a equality. We will do that later for Fourier series

12 Fuctioals Defiitio: Cosider, where is a Hilert space. If to each x, there is a complex umer T [x], the T is a fuctioal o. A fuctioal is ouded o if there is a costat c such that for all x, T [x] c x. The smallest c for which this iequality holds is called the orm of T ad is writte T. A fuctioal is cotiuous if for every x ad for every sequece {x } x, we have T[x ] T[x]. Defiitio: A fuctioal is liear if T [x 1 + x ] = T [x 1 ] + T [x ]; T [x] = T[x] Remar: T[0] = 0; T jxj jt x j (for fiite) j1 j1 Remar: T[x] ] = x, y is a liear fuctioal. Whe represeted dy matrices, we would have Tx = y T x. Note that the row vector is represetig a fuctioal, ot a astract vector. Note that y, x is ot liear; it is sometimes called ati-liear 6.3 Fuctioals Theorem 1: If a liear fuctioal is cotiuous at x = 0, the it is cotiuous o the etire domai of defiitio D T Theorem : A fuctioal is ouded if ad oly if it is cotiuous Theorem 3 (Riesz represetatio ti theorem): Each cotiuous liear fuctioal T[x] o a Hilert space ca e expressed i the form x, f, where f is a fixed elemet i. Proof: Cosider the set of vectors i the ull space of T[x]. Either =, i which case we set f = 0, or dim 1, ad we may fid a elemet f 0 such that f 0 = 1 i. We ow set f T[ f. Let y = T[x] f 0 xt[ f 0 ]. The 0] f0 T[ y] = 0 for all x, ad y. It follows that Txf [ ] 0 xt [ f0 ], f0 Tx [ ] T [ f0 ] xf, 0 so that Tx [ ] T[ f0] xf, 0 xt, [ f0] f0 To demostrate uiqueess, we ote that if x, f = x, g for all x, the x, f g = 0 ad usig x = f g, we coclude f g. Remar: We ifer dim = 0 or

13 Fuctioals Theorem 4: Every liear fuctioal i is ouded ( ) ( ) Proof: It suffices to show that limt x 0 for every { x } 0 0 We tae a orthoormal asis { 1,,, }, ad we write ( ) ( ) ( ) ( ) x j j, where j x, j j1 ( ) j 0 ( ) ( ) limt x lim j T j 0 j 1 From cotiuity ad {x () } 0, we ifer. Hece, Remar: This theorem does ot hold for liear fuctioals i. The classic couter-example example is the -fuctio: T[x] = x(0). Taig, 1/ t 1/ x () t 0, otherwise we see that x (0) / x as 6.5 Fuctioals Dual Bases Let {e 1,, e } e a fixed aritrary asis i. The effect of T o ay vector x is completely descried y complex umers T[e 1 ],, T[e ] sice TxT jej jt e j j1 j1 Coversely, give T[e 1 ] = 1,, T[e ] =, there is exactly oe liear fuctioal that it defies. I particular, if we let T [e j ] = j, the the Riesz theorem tells us that there is a uique vector e* that satisfies the relatio ej, e* j The set of vectors { e1*, e*,..., e*} is called the dual or reciprocal asis. (How do we ow that this set of vectors is a asis?) Remar: Dual ases are very importat i lossy (o-hermitia) prolems, where it is ofte coveiet to use a o-orthoormal asis sice we may write

14 Dual Bases Fuctioals Remar: Dual ases are very importat i lossy (o-hermitia) prolems, where it is ofte coveiet to use a o-orthoormal asis sice we may write x e, where x, e* j1 j j j j Developig ad usig dual ases i large systems remais a active area of research! 6.7 Trasformatios Defiitio: A trasformatio A: is a fuctio that maps x ito y, where,. The domai, the rage, ad the ull space of the trasformatio A are deoted A (= ), A, ad A respectively Examples: (1) x = ( 1,,, m ) y = m : I this case, we may choose = m, ad we fid =, A =, A = (x : x 0), A = {0} () x( 1,,, : I this case, we may choose = m m) y x ( 1,,, m), ad we fid = A = A = m, ad A = {0} (3) The Fourier trasform: y( f ) x( t)exp( ift) dt I this case, we may choose = (, ), ad we fid = A = A = (, ), ad A = {0}

15 Trasformatios Defiitio: A trasformatio A: is liear if A (x 1 + x ) = Ax 1 + Ax ; A (x) = A (x) Remar: I this case, A, A, ad A are all liear maifolds Remar: All liear trasformatios from m ca e represeted as matrices ad vice versa. Liear trasformatios from/to ifiite-dimesioal systems are also represeted as matrices whe discretized for computatio. Defiitio: A trasformatio is ouded o its domai if for all x A, there exists a costat c such that Ax c x. We the defie the orm of A, A, as the lowest upper oud of Ax / x. Theorems: As i the case of ffuctioals, we fid: (1) If a liear trasformatio ti is cotiuous at x = 0, it is cotiuous o all of A. () A liear trasformatio is cotiuous if ad oly if it is ouded. (3) All liear trasformatios m are ouded. 6.9 Trasformatios Adjoit Trasformatios i ( from ) Defiitio: Let A e a liear trasformatio ad f a fixed vector i. The Ax, f is a liear fuctioal that must e cotiuous. From the Riesz theorem, we ow there is some g such that Ax, f = x, g. We may write g = A*f, where we fid that A* is a liear trasformatio. The trasformatio A* is the adjoit of A. For every x ad y, we have Ax, y = x, A*y. The correspodig matrices A = [a j ] ad A* = [ a* j ] are Hermitia cojugates with respect to a orthoormal asis. Alterative Theorem: The equatio Ax = f has solutios if ad oly if f, z = 0 for every solutio that is a solutio of A*z = 0, i.e., f A if ad oly if f is orthogoal to the ull space of A*; that is A = ( A* )

16 Adjoit Trasformatios i Trasformatios Alterative Theorem: The equatio Ax = f has solutios if ad oly if f, z = 0 for every solutio that is a solutio of A*z = 0, i.e., f A if ad oly if f is orthogoal to the ull space of A*; * that is A = ( A* ) Proof: We will prove A* = ( A ), which is equivalet sice () = for ay liear maifold. (a) We first show ( A ) A* : Let z e i ( A ) ; the f, z = 0 for every f A. Sice the domai equals the whole space, we have for every x, Ax, z = 0. Hece, x, A*z = 0 for all x ad A*z = 0. () We ow show A* ( A ) : Let z A* ad f A. Hece, A*z = 0 ad there exists a x such that Ax = f. The, f, z = Ax, z = x, A*z = x, 0 = 0 Corollary: A = ( A* ). We ote (A*)* = A Theorem: ra A = ra A*; ullity A = ullity A*. Proof: We have ra A + ullity A* = ra A* + ullity A =. We earlier showed usig matrices that ra A + ullity A =. The result follows

Chapter 3 Inner Product Spaces. Hilbert Spaces

Chapter 3 Inner Product Spaces. Hilbert Spaces Chapter 3 Ier Product Spaces. Hilbert Spaces 3. Ier Product Spaces. Hilbert Spaces 3.- Defiitio. A ier product space is a vector space X with a ier product defied o X. A Hilbert space is a complete ier

More information

Introduction to Optimization Techniques

Introduction to Optimization Techniques Itroductio to Optimizatio Techiques Basic Cocepts of Aalysis - Real Aalysis, Fuctioal Aalysis 1 Basic Cocepts of Aalysis Liear Vector Spaces Defiitio: A vector space X is a set of elemets called vectors

More information

Definition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.

Definition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4. 4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad

More information

Math Solutions to homework 6

Math Solutions to homework 6 Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there

More information

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3 MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special

More information

Archimedes - numbers for counting, otherwise lengths, areas, etc. Kepler - geometry for planetary motion

Archimedes - numbers for counting, otherwise lengths, areas, etc. Kepler - geometry for planetary motion Topics i Aalysis 3460:589 Summer 007 Itroductio Ree descartes - aalysis (breaig dow) ad sythesis Sciece as models of ature : explaatory, parsimoious, predictive Most predictios require umerical values,

More information

LECTURE 8: ORTHOGONALITY (CHAPTER 5 IN THE BOOK)

LECTURE 8: ORTHOGONALITY (CHAPTER 5 IN THE BOOK) LECTURE 8: ORTHOGONALITY (CHAPTER 5 IN THE BOOK) Everythig marked by is ot required by the course syllabus I this lecture, all vector spaces is over the real umber R. All vectors i R is viewed as a colum

More information

A) is empty. B) is a finite set. C) can be a countably infinite set. D) can be an uncountable set.

A) is empty. B) is a finite set. C) can be a countably infinite set. D) can be an uncountable set. M.A./M.Sc. (Mathematics) Etrace Examiatio 016-17 Max Time: hours Max Marks: 150 Istructios: There are 50 questios. Every questio has four choices of which exactly oe is correct. For correct aswer, 3 marks

More information

Real Numbers R ) - LUB(B) may or may not belong to B. (Ex; B= { y: y = 1 x, - Note that A B LUB( A) LUB( B)

Real Numbers R ) - LUB(B) may or may not belong to B. (Ex; B= { y: y = 1 x, - Note that A B LUB( A) LUB( B) Real Numbers The least upper boud - Let B be ay subset of R B is bouded above if there is a k R such that x k for all x B - A real umber, k R is a uique least upper boud of B, ie k = LUB(B), if () k is

More information

We are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n

We are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at

More information

Singular Continuous Measures by Michael Pejic 5/14/10

Singular Continuous Measures by Michael Pejic 5/14/10 Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable

More information

2 Banach spaces and Hilbert spaces

2 Banach spaces and Hilbert spaces 2 Baach spaces ad Hilbert spaces Tryig to do aalysis i the ratioal umbers is difficult for example cosider the set {x Q : x 2 2}. This set is o-empty ad bouded above but does ot have a least upper boud

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

More information

Chapter 6 Infinite Series

Chapter 6 Infinite Series Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat

More information

Introduction to Functional Analysis

Introduction to Functional Analysis MIT OpeCourseWare http://ocw.mit.edu 18.10 Itroductio to Fuctioal Aalysis Sprig 009 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. LECTURE OTES FOR 18.10,

More information

Sequence A sequence is a function whose domain of definition is the set of natural numbers.

Sequence A sequence is a function whose domain of definition is the set of natural numbers. Chapter Sequeces Course Title: Real Aalysis Course Code: MTH3 Course istructor: Dr Atiq ur Rehma Class: MSc-I Course URL: wwwmathcityorg/atiq/fa8-mth3 Sequeces form a importat compoet of Mathematical Aalysis

More information

Introduction to Optimization Techniques. How to Solve Equations

Introduction to Optimization Techniques. How to Solve Equations Itroductio to Optimizatio Techiques How to Solve Equatios Iterative Methods of Optimizatio Iterative methods of optimizatio Solutio of the oliear equatios resultig form a optimizatio problem is usually

More information

Math 61CM - Solutions to homework 3

Math 61CM - Solutions to homework 3 Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig

More information

Mathematical Methods for Physics and Engineering

Mathematical Methods for Physics and Engineering Mathematical Methods for Physics ad Egieerig Lecture otes Sergei V. Shabaov Departmet of Mathematics, Uiversity of Florida, Gaiesville, FL 326 USA CHAPTER The theory of covergece. Numerical sequeces..

More information

PAPER : IIT-JAM 2010

PAPER : IIT-JAM 2010 MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure

More information

1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.

1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series. .3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(

More information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information

Ma 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5

Ma 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5 Ma 42: Itroductio to Lebesgue Itegratio Solutios to Homework Assigmet 5 Prof. Wickerhauser Due Thursday, April th, 23 Please retur your solutios to the istructor by the ed of class o the due date. You

More information

Axioms of Measure Theory

Axioms of Measure Theory MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that

More information

Solutions to home assignments (sketches)

Solutions to home assignments (sketches) Matematiska Istitutioe Peter Kumli 26th May 2004 TMA401 Fuctioal Aalysis MAN670 Applied Fuctioal Aalysis 4th quarter 2003/2004 All documet cocerig the course ca be foud o the course home page: http://www.math.chalmers.se/math/grudutb/cth/tma401/

More information

lim za n n = z lim a n n.

lim za n n = z lim a n n. Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget

More information

} is said to be a Cauchy sequence provided the following condition is true.

} is said to be a Cauchy sequence provided the following condition is true. Math 4200, Fial Exam Review I. Itroductio to Proofs 1. Prove the Pythagorea theorem. 2. Show that 43 is a irratioal umber. II. Itroductio to Logic 1. Costruct a truth table for the statemet ( p ad ~ r

More information

Lecture 3 The Lebesgue Integral

Lecture 3 The Lebesgue Integral Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified

More information

3 Gauss map and continued fractions

3 Gauss map and continued fractions ICTP, Trieste, July 08 Gauss map ad cotiued fractios I this lecture we will itroduce the Gauss map, which is very importat for its coectio with cotiued fractios i umber theory. The Gauss map G : [0, ]

More information

Riesz-Fischer Sequences and Lower Frame Bounds

Riesz-Fischer Sequences and Lower Frame Bounds Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 1 (00), No., 305 314 Riesz-Fischer Sequeces ad Lower Frame Bouds P. Casazza, O. Christese, S. Li ad A. Lider Abstract.

More information

REAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS

REAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS REAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS 18th Feb, 016 Defiitio (Lipschitz fuctio). A fuctio f : R R is said to be Lipschitz if there exists a positive real umber c such that for ay x, y i the domai

More information

7 Sequences of real numbers

7 Sequences of real numbers 40 7 Sequeces of real umbers 7. Defiitios ad examples Defiitio 7... A sequece of real umbers is a real fuctio whose domai is the set N of atural umbers. Let s : N R be a sequece. The the values of s are

More information

CHAPTER 5. Theory and Solution Using Matrix Techniques

CHAPTER 5. Theory and Solution Using Matrix Techniques A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL

More information

Review Problems 1. ICME and MS&E Refresher Course September 19, 2011 B = C = AB = A = A 2 = A 3... C 2 = C 3 = =

Review Problems 1. ICME and MS&E Refresher Course September 19, 2011 B = C = AB = A = A 2 = A 3... C 2 = C 3 = = Review Problems ICME ad MS&E Refresher Course September 9, 0 Warm-up problems. For the followig matrices A = 0 B = C = AB = 0 fid all powers A,A 3,(which is A times A),... ad B,B 3,... ad C,C 3,... Solutio:

More information

Ma 530 Infinite Series I

Ma 530 Infinite Series I Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li

More information

MATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and

MATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and MATH01 Real Aalysis (2008 Fall) Tutorial Note #7 Sequece ad Series of fuctio 1: Poitwise Covergece ad Uiform Covergece Part I: Poitwise Covergece Defiitio of poitwise covergece: A sequece of fuctios f

More information

Ma 530 Introduction to Power Series

Ma 530 Introduction to Power Series Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power

More information

Math 113, Calculus II Winter 2007 Final Exam Solutions

Math 113, Calculus II Winter 2007 Final Exam Solutions Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this

More information

Product measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.

Product measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014. Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the

More information

Filter banks. Separately, the lowpass and highpass filters are not invertible. removes the highest frequency 1/ 2and

Filter banks. Separately, the lowpass and highpass filters are not invertible. removes the highest frequency 1/ 2and Filter bas Separately, the lowpass ad highpass filters are ot ivertible T removes the highest frequecy / ad removes the lowest frequecy Together these filters separate the sigal ito low-frequecy ad high-frequecy

More information

TENSOR PRODUCTS AND PARTIAL TRACES

TENSOR PRODUCTS AND PARTIAL TRACES Lecture 2 TENSOR PRODUCTS AND PARTIAL TRACES Stéphae ATTAL Abstract This lecture cocers special aspects of Operator Theory which are of much use i Quatum Mechaics, i particular i the theory of Quatum Ope

More information

(VII.A) Review of Orthogonality

(VII.A) Review of Orthogonality VII.A Review of Orthogoality At the begiig of our study of liear trasformatios i we briefly discussed projectios, rotatios ad projectios. I III.A, projectios were treated i the abstract ad without regard

More information

Chapter 8. Uniform Convergence and Differentiation.

Chapter 8. Uniform Convergence and Differentiation. Chapter 8 Uiform Covergece ad Differetiatio This chapter cotiues the study of the cosequece of uiform covergece of a series of fuctio I Chapter 7 we have observed that the uiform limit of a sequece of

More information

Fall 2013 MTH431/531 Real analysis Section Notes

Fall 2013 MTH431/531 Real analysis Section Notes Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters

More information

Lecture Notes for Analysis Class

Lecture Notes for Analysis Class Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios

More information

Theorem 3. A subset S of a topological space X is compact if and only if every open cover of S by open sets in X has a finite subcover.

Theorem 3. A subset S of a topological space X is compact if and only if every open cover of S by open sets in X has a finite subcover. Compactess Defiitio 1. A cover or a coverig of a topological space X is a family C of subsets of X whose uio is X. A subcover of a cover C is a subfamily of C which is a cover of X. A ope cover of X is

More information

University of Colorado Denver Dept. Math. & Stat. Sciences Applied Analysis Preliminary Exam 13 January 2012, 10:00 am 2:00 pm. Good luck!

University of Colorado Denver Dept. Math. & Stat. Sciences Applied Analysis Preliminary Exam 13 January 2012, 10:00 am 2:00 pm. Good luck! Uiversity of Colorado Dever Dept. Math. & Stat. Scieces Applied Aalysis Prelimiary Exam 13 Jauary 01, 10:00 am :00 pm Name: The proctor will let you read the followig coditios before the exam begis, ad

More information

[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.

[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms. [ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural

More information

Chimica Inorganica 3

Chimica Inorganica 3 himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule

More information

EE / EEE SAMPLE STUDY MATERIAL. GATE, IES & PSUs Signal System. Electrical Engineering. Postal Correspondence Course

EE / EEE SAMPLE STUDY MATERIAL. GATE, IES & PSUs Signal System. Electrical Engineering. Postal Correspondence Course Sigal-EE Postal Correspodece Course 1 SAMPLE STUDY MATERIAL Electrical Egieerig EE / EEE Postal Correspodece Course GATE, IES & PSUs Sigal System Sigal-EE Postal Correspodece Course CONTENTS 1. SIGNAL

More information

Homework 2. Show that if h is a bounded sesquilinear form on the Hilbert spaces X and Y, then h has the representation

Homework 2. Show that if h is a bounded sesquilinear form on the Hilbert spaces X and Y, then h has the representation omework 2 1 Let X ad Y be ilbert spaces over C The a sesquiliear form h o X Y is a mappig h : X Y C such that for all x 1, x 2, x X, y 1, y 2, y Y ad all scalars α, β C we have (a) h(x 1 + x 2, y) h(x

More information

If a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?

If a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero? 2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a

More information

6. Uniform distribution mod 1

6. Uniform distribution mod 1 6. Uiform distributio mod 1 6.1 Uiform distributio ad Weyl s criterio Let x be a seuece of real umbers. We may decompose x as the sum of its iteger part [x ] = sup{m Z m x } (i.e. the largest iteger which

More information

Advanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology

Advanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology Advaced Aalysis Mi Ya Departmet of Mathematics Hog Kog Uiversity of Sciece ad Techology September 3, 009 Cotets Limit ad Cotiuity 7 Limit of Sequece 8 Defiitio 8 Property 3 3 Ifiity ad Ifiitesimal 8 4

More information

IIT JAM Mathematical Statistics (MS) 2006 SECTION A

IIT JAM Mathematical Statistics (MS) 2006 SECTION A IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim

More information

Continuous Functions

Continuous Functions Cotiuous Fuctios Q What does it mea for a fuctio to be cotiuous at a poit? Aswer- I mathematics, we have a defiitio that cosists of three cocepts that are liked i a special way Cosider the followig defiitio

More information

Topics. Homework Problems. MATH 301 Introduction to Analysis Chapter Four Sequences. 1. Definition of convergence of sequences.

Topics. Homework Problems. MATH 301 Introduction to Analysis Chapter Four Sequences. 1. Definition of convergence of sequences. MATH 301 Itroductio to Aalysis Chapter Four Sequeces Topics 1. Defiitio of covergece of sequeces. 2. Fidig ad provig the limit of sequeces. 3. Bouded covergece theorem: Theorem 4.1.8. 4. Theorems 4.1.13

More information

MA131 - Analysis 1. Workbook 3 Sequences II

MA131 - Analysis 1. Workbook 3 Sequences II MA3 - Aalysis Workbook 3 Sequeces II Autum 2004 Cotets 2.8 Coverget Sequeces........................ 2.9 Algebra of Limits......................... 2 2.0 Further Useful Results........................

More information

CHAPTER 3. GOE and GUE

CHAPTER 3. GOE and GUE CHAPTER 3 GOE ad GUE We quicly recall that a GUE matrix ca be defied i the followig three equivalet ways. We leave it to the reader to mae the three aalogous statemets for GOE. I the previous chapters,

More information

Math 220A Fall 2007 Homework #2. Will Garner A

Math 220A Fall 2007 Homework #2. Will Garner A Math 0A Fall 007 Homewor # Will Garer Pg 3 #: Show that {cis : a o-egative iteger} is dese i T = {z œ : z = }. For which values of q is {cis(q): a o-egative iteger} dese i T? To show that {cis : a o-egative

More information

1 1 2 = show that: over variables x and y. [2 marks] Write down necessary conditions involving first and second-order partial derivatives for ( x0, y

1 1 2 = show that: over variables x and y. [2 marks] Write down necessary conditions involving first and second-order partial derivatives for ( x0, y Questio (a) A square matrix A= A is called positive defiite if the quadratic form waw > 0 for every o-zero vector w [Note: Here (.) deotes the traspose of a matrix or a vector]. Let 0 A = 0 = show that:

More information

Module 18 Discrete Time Signals and Z-Transforms Objective: Introduction : Description: Discrete Time Signal representation

Module 18 Discrete Time Signals and Z-Transforms Objective: Introduction : Description: Discrete Time Signal representation Module 8 Discrete Time Sigals ad Z-Trasforms Objective:To uderstad represetig discrete time sigals, apply z trasform for aalyzigdiscrete time sigals ad to uderstad the relatio to Fourier trasform Itroductio

More information

Functional Analysis I

Functional Analysis I Fuctioal Aalysis I Term 1, 2009 2010 Vassili Gelfreich Cotets 1 Vector spaces 1 1.1 Defiitio................................. 1 1.2 Examples of vector spaces....................... 2 1.3 Hamel bases...............................

More information

R is a scalar defined as follows:

R is a scalar defined as follows: Math 8. Notes o Dot Product, Cross Product, Plaes, Area, ad Volumes This lecture focuses primarily o the dot product ad its may applicatios, especially i the measuremet of agles ad scalar projectio ad

More information

The Discrete Fourier Transform

The Discrete Fourier Transform The Discrete Fourier Trasform Complex Fourier Series Represetatio Recall that a Fourier series has the form a 0 + a k cos(kt) + k=1 b k si(kt) This represetatio seems a bit awkward, sice it ivolves two

More information

Beurling Integers: Part 2

Beurling Integers: Part 2 Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers

More information

, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx)

, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx) Cosider the differetial equatio y '' k y 0 has particular solutios y1 si( kx) ad y cos( kx) I geeral, ay liear combiatio of y1 ad y, cy 1 1 cy where c1, c is also a solutio to the equatio above The reaso

More information

The second is the wish that if f is a reasonably nice function in E and φ n

The second is the wish that if f is a reasonably nice function in E and φ n 8 Sectio : Approximatios i Reproducig Kerel Hilbert Spaces I this sectio, we address two cocepts. Oe is the wish that if {E, } is a ierproduct space of real valued fuctios o the iterval [,], the there

More information

Introduction to Extreme Value Theory Laurens de Haan, ISM Japan, Erasmus University Rotterdam, NL University of Lisbon, PT

Introduction to Extreme Value Theory Laurens de Haan, ISM Japan, Erasmus University Rotterdam, NL University of Lisbon, PT Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Itroductio to Extreme Value Theory Laures de Haa Erasmus Uiversity Rotterdam, NL Uiversity of Lisbo, PT Itroductio to Extreme Value Theory

More information

6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.

6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer. 6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio

More information

Apply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.

Apply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j. Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α

More information

Chapter 7: The z-transform. Chih-Wei Liu

Chapter 7: The z-transform. Chih-Wei Liu Chapter 7: The -Trasform Chih-Wei Liu Outlie Itroductio The -Trasform Properties of the Regio of Covergece Properties of the -Trasform Iversio of the -Trasform The Trasfer Fuctio Causality ad Stability

More information

Singular value decomposition. Mathématiques appliquées (MATH0504-1) B. Dewals, Ch. Geuzaine

Singular value decomposition. Mathématiques appliquées (MATH0504-1) B. Dewals, Ch. Geuzaine Lecture 11 Sigular value decompositio Mathématiques appliquées (MATH0504-1) B. Dewals, Ch. Geuzaie V1.2 07/12/2018 1 Sigular value decompositio (SVD) at a glace Motivatio: the image of the uit sphere S

More information

L = n i, i=1. dp p n 1

L = n i, i=1. dp p n 1 Exchageable sequeces ad probabilities for probabilities 1996; modified 98 5 21 to add material o mutual iformatio; modified 98 7 21 to add Heath-Sudderth proof of de Fietti represetatio; modified 99 11

More information

MAT1026 Calculus II Basic Convergence Tests for Series

MAT1026 Calculus II Basic Convergence Tests for Series MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real

More information

Inverse Matrix. A meaning that matrix B is an inverse of matrix A.

Inverse Matrix. A meaning that matrix B is an inverse of matrix A. Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix

More information

Continued Fractions and Pell s Equation

Continued Fractions and Pell s Equation Max Lah Joatha Spiegel May, 06 Abstract Cotiued fractios provide a useful, ad arguably more atural, way to uderstad ad represet real umbers as a alterative to decimal expasios I this paper, we eumerate

More information

MA131 - Analysis 1. Workbook 9 Series III

MA131 - Analysis 1. Workbook 9 Series III MA3 - Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................

More information

Seunghee Ye Ma 8: Week 5 Oct 28

Seunghee Ye Ma 8: Week 5 Oct 28 Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value

More information

Assignment 1 : Real Numbers, Sequences. for n 1. Show that (x n ) converges. Further, by observing that x n+2 + x n+1

Assignment 1 : Real Numbers, Sequences. for n 1. Show that (x n ) converges. Further, by observing that x n+2 + x n+1 Assigmet : Real Numbers, Sequeces. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a upper boud of A for every N. 2. Let y (, ) ad x (, ). Evaluate

More information

Chapter 7 z-transform

Chapter 7 z-transform Chapter 7 -Trasform Itroductio Trasform Uilateral Trasform Properties Uilateral Trasform Iversio of Uilateral Trasform Determiig the Frequecy Respose from Poles ad Zeros Itroductio Role i Discrete-Time

More information

Final Solutions. 1. (25pts) Define the following terms. Be as precise as you can.

Final Solutions. 1. (25pts) Define the following terms. Be as precise as you can. Mathematics H104 A. Ogus Fall, 004 Fial Solutios 1. (5ts) Defie the followig terms. Be as recise as you ca. (a) (3ts) A ucoutable set. A ucoutable set is a set which ca ot be ut ito bijectio with a fiite

More information

8. Applications To Linear Differential Equations

8. Applications To Linear Differential Equations 8. Applicatios To Liear Differetial Equatios 8.. Itroductio 8.. Review Of Results Cocerig Liear Differetial Equatios Of First Ad Secod Orders 8.3. Eercises 8.4. Liear Differetial Equatios Of Order N 8.5.

More information

Machine Learning for Data Science (CS 4786)

Machine Learning for Data Science (CS 4786) Machie Learig for Data Sciece CS 4786) Lecture & 3: Pricipal Compoet Aalysis The text i black outlies high level ideas. The text i blue provides simple mathematical details to derive or get to the algorithm

More information

Integrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number

Integrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios

More information

Chapter 2 The Monte Carlo Method

Chapter 2 The Monte Carlo Method Chapter 2 The Mote Carlo Method The Mote Carlo Method stads for a broad class of computatioal algorithms that rely o radom sampligs. It is ofte used i physical ad mathematical problems ad is most useful

More information

Physics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.

Physics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001. Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;

More information

Math 113 Exam 3 Practice

Math 113 Exam 3 Practice Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you

More information

5 Birkhoff s Ergodic Theorem

5 Birkhoff s Ergodic Theorem 5 Birkhoff s Ergodic Theorem Amog the most useful of the various geeralizatios of KolmogorovâĂŹs strog law of large umbers are the ergodic theorems of Birkhoff ad Kigma, which exted the validity of the

More information

1+x 1 + α+x. x = 2(α x2 ) 1+x

1+x 1 + α+x. x = 2(α x2 ) 1+x Math 2030 Homework 6 Solutios # [Problem 5] For coveiece we let α lim sup a ad β lim sup b. Without loss of geerality let us assume that α β. If α the by assumptio β < so i this case α + β. By Theorem

More information

Appendix: The Laplace Transform

Appendix: The Laplace Transform Appedix: The Laplace Trasform The Laplace trasform is a powerful method that ca be used to solve differetial equatio, ad other mathematical problems. Its stregth lies i the fact that it allows the trasformatio

More information

ENGI Series Page 6-01

ENGI Series Page 6-01 ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,

More information

FUNDAMENTALS OF REAL ANALYSIS by

FUNDAMENTALS OF REAL ANALYSIS by FUNDAMENTALS OF REAL ANALYSIS by Doğa Çömez Backgroud: All of Math 450/1 material. Namely: basic set theory, relatios ad PMI, structure of N, Z, Q ad R, basic properties of (cotiuous ad differetiable)

More information

A Note on the Symmetric Powers of the Standard Representation of S n

A Note on the Symmetric Powers of the Standard Representation of S n A Note o the Symmetric Powers of the Stadard Represetatio of S David Savitt 1 Departmet of Mathematics, Harvard Uiversity Cambridge, MA 0138, USA dsavitt@mathharvardedu Richard P Staley Departmet of Mathematics,

More information

MA541 : Real Analysis. Tutorial and Practice Problems - 1 Hints and Solutions

MA541 : Real Analysis. Tutorial and Practice Problems - 1 Hints and Solutions MA54 : Real Aalysis Tutorial ad Practice Problems - Hits ad Solutios. Suppose that S is a oempty subset of real umbers that is bouded (i.e. bouded above as well as below). Prove that if S sup S. What ca

More information

MTH Assignment 1 : Real Numbers, Sequences

MTH Assignment 1 : Real Numbers, Sequences MTH -26 Assigmet : Real Numbers, Sequeces. Fid the supremum of the set { m m+ : N, m Z}. 2. Let A be a o-empty subset of R ad α R. Show that α = supa if ad oly if α is ot a upper boud of A but α + is a

More information

DANIELL AND RIEMANN INTEGRABILITY

DANIELL AND RIEMANN INTEGRABILITY DANIELL AND RIEMANN INTEGRABILITY ILEANA BUCUR We itroduce the otio of Riema itegrable fuctio with respect to a Daiell itegral ad prove the approximatio theorem of such fuctios by a mootoe sequece of Jorda

More information

f(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim

f(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =

More information

The Gamma function Michael Taylor. Abstract. This material is excerpted from 18 and Appendix J of [T].

The Gamma function Michael Taylor. Abstract. This material is excerpted from 18 and Appendix J of [T]. The Gamma fuctio Michael Taylor Abstract. This material is excerpted from 8 ad Appedix J of [T]. The Gamma fuctio has bee previewed i 5.7 5.8, arisig i the computatio of a atural Laplace trasform: 8. ft

More information