Advanced Linear Algebra Lecture Notes

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Math 412: Advanced Linear Algebra Lecture Notes Lior SilbermanThese are rough notes for the Spring 2014 course. Solutions to problem sets were posted on an internal website.Contents Introduction 5 0.1. Administrivia 5 0.2. Euler’s Theorem 5 0.3. Course plan (subject to revision) (Lecture 1, 6/1/13) 6 0.4. Review 7 Math 412: Problem Set 1 (due 15/1/2014) 8 Chapter 1. Constructions 11 1.1. Direct sum, direct product 11 1.2. Quotients (Lecture 5, 15/1/2015) 12 1.3. Hom spaces and duality 13 Math 412: Problem Set 2 (due 22/1/2014) 14 Math 412: Problem Set 3 (due 29/1/2014) 20 1.4. Multilinear algebra and tensor products 22 Math 412: Problem Set 4 (due 7/2/2014) 24 Math 412: Supplementary Problem Set on Categories 26 Math 412: Problem Set 5 (due 14/2/2014) 30 Chapter 2. Structure Theory 32 2.1. Introduction (Lecture 15,7/2/14) 32 2.2. Jordan Canonical Form 33 Math 412: Problem Set 6 (due 28/2/2014) 43 Math 412: Problem set 7 (due 10/3/2014) 45 Chapter 3. Vector and matrix norms 47 3.1. Review of metric spaces 47 3.2. Norms on vector spaces 47 3.3. Norms on matrices 49 Math 412: Problem set 8, due 19/3/2014 51 3.4. Example: eigenvalues and the power method (Lecture, 17/ 54 3.5. Sequences and series of vectors and matrices 54 Math 412: Problem set 9, due 26/3/2014 57 Chapter 4. The Holomorphic Calculus 58 4.1. The exponential series (24/3/2014) 58 4.2. 26/3/2014 60 Math 412: Problem set 10, due 7/4/2014 61 4.3. Invertibility and the resolvent (31/3/2014) 64 3Chapter 5. Vignettes 65 5.1. The exponential map and structure theory for GL (R) (2/4/2014) 65 n 5.2. Representation Theory of Groups 65 Bibliography 66 4Introduction Lior Silberman,liorMath.UBC.CA, Office: Math Building 229B Phone: 604-827-3031 0.1. Administrivia  Problem sets will be posted on the course website. – To the extent I have time, solutions may be posted on Connect. – If you create solution sets I’ll certainly post them.  Textbooks – Halmos – Algebra books 0.2. Euler’s Theorem Let G=(V;E) be a connected planar graph. A face of G is a finite connected component of 2 R n G. THEOREM 1 (Euler). v e+ f = 1. E V PROOF. Arbitrarily orient the edges. Let ¶ :R R be defined by f((u;v))= 1 1 , E v u F E ¶ :R R be given by the sum of edges around the face. F LEMMA 2. ¶ is injective. F PROOF. Faces containing boundary edges are independent. Remove them and repeat.  LEMMA 3. Ker¶ = Im¶ . E F PROOF. Suppose a combo of edges is in the kernel. Following a sequence with non-zero coefficients gives a closed loop, which can be expressed as a sum of faces. Now subtract a multiple to reduce the number of edges with non-zero coefficients.  LEMMA 4. Im(¶ ) is the the set of functions with total weight zero. E PROOF. Clearly the image is contained there. Conversely, given f of total weight zero move the weight to a single vertex using elements of the image. remark: quotient vector spaces  E Now dimR = dimKer¶ + dimIm¶ = dimIm¶ + dimIm¶ so E E F E e= f+(v 1):  REMARK 5. UsingF coefficients is even simpler. 2 50.3. Course plan (subject to revision) (Lecture 1, 6/1/13)  Quick review  Some constructions – Direct sum and product – Spaces of homomorphisms and duality – Quotient vector spaces – Multilinear algebra: Tensor products  Structure theory for linear maps – Gram–Schmidt, Polar, Cartan † – The Bruhat decompositions and LU, LL factorization; numerical applications – The minimal polynomial and the Cayley–Hamilton Theorem – The Jordan canonical form  Analysis with vectors and matrices – Norms on vector spaces – Operator norms tX – Matrices in power series: e and its friends.  Other topics if time permits. 60.4. Review 0.4.1. Basic definitions. We want to give ourselves the freedom to have scalars other than real or complex. DEFINITION 6 (Fields). A field is a quintuple(F;0;1;+;) such that(F;0;+) and(Fnf0g;1;) are abelian groups, and the distributive law8x;y;z2 F : x(y+ z)= xy+ xz holds. EXAMPLE 7. R,C,Q.F (via addition and multiplication tables; ex: show this is a field),F . 2 p r EXERCISE 8. Every finite field has p elements for some prime p and some integer r 1. Fact: there is one such field for every prime power. DEFINITION 9. A vector space over a field F is a quadruple (V;0;+;) where (V;0;+) is an abelian group, and: FV V is a map such that: (1) 1 v= v. F (2) a(bv)=(ab)v. (3) (a+b)(v+ w)=av+bv+aw+bw. LEMMA 10. 0  v= 0 for all v. F PROOF. 0v=(0+ 0)v= 0v+ 0v. Now subtract 0v from both sides.  0.4.2. Bases and dimension. Fix a vector space V . DEFINITION 11. Let S V . r r r  v2 V depends on S if there arefvg  S andfag  F such that v= a v empty å i i i i=1 i i=1 i=1 sum is 0  Write Span (S) V for the set of vectors that depend on S. F  Call S linearly dependent if some v2 S depends on Snfvg, equivalently if there are r r r distinctfvg  S andfag  F such that a v = 0. å i i i i=1 i=1 i i=1  Call S linearly independent if it is not linearly dependent. AXIOM 12 (Axiom of choice). Every vector space has a basis. n X 0.4.3. Examples.f0g,R , F . 7Math 412: Problem Set 1 (due 15/1/2014) Practice problems, any sub-parts marked “OPT” (optional) and supplementary problems are not for submission. Practice problems 3 P1 Show that the map f :R R given by f(x;y;z)= x 2y+ z is a linear map. Show that the 2 maps(x;y;z)7 1 and(x;y;z)7 x are non-linear. P2 Let F be a field, X a set. Carefully show that pointwise addition and scalar multiplication X endow the set F of functions from X to F with the structure of an F-vectorspace. For submission RMK The following idea will be used repeatedly during the course to prove that sets of vectors are linearly independent. Make sure you understand how this argument works. 1. Let V be a vector space, S V a set of vectors. A minimal dependence in S is an equality m a v = 0 where v2 S are distinct, a are scalars not all of which are zero, and m 1 is as å i i i=1 i i small as possible so that suchfag,fvg exist. i i 80 1 0 1 0 1 0 19 1 1 1 2 = 3 A A A A (a) Find a minimal dependence among 1 ; 0 ; 1 ; 1 R . : ; 0 1 1 1 (b) Show that in a minimal dependence the a are all non-zero. i m m (c) Suppose that a v and b v are minimal dependences in S, involving the exact å å i i i=1 i i=1 i same set of vectors. Show that there is a non-zero scalar c such that a = cb . i i (d) Let T : V V be a linear map, and let S V be a set of (non-zero) eigenvectors of T , each corresponding to a distinct eigenvalue. Applying T to a minimal dependence in S obtain a contradiction to (b) and conclude that S is actually linearly independent.  (e) LetG be a group. The set Hom(G;C ) of group homomorphisms fromG to the multi- plicative group of nonzero complex numbers is called the set of quasicharacters ofG (the notion of “character of a group” has an additional, different but related meaning, which is  not at issue in this problem). Show that Hom(G;C ) is linearly independent in the space G C of functions fromG toC. ¥ ¥ 2. Let S=fcos(nx)g fsin(nx)g , thought of as a subset of the space C(p;p) of contin- n=0 n=1 uous functions on the intervalp;p. d (a) Applying to a putative minimal dependence in S obtain a different linear dependence dx of at most the same length, and use that to show that S is, in fact, linearly independent. (b) Show that the elements of S are an orthogonal system with respect to the inner product R p h f;gi= f(x)g(x)dx (feel free to look up any trig identities you need). This gives a p different proof of their independence. (c) Let W = Span (S) (this is usually called “the space of trigonometric polynomials”; a C p typical element is 5 sin(3x)+ 2cos(15x)p cos(32x)). Find a ordering of S so that d the matrix of the linear map : W W in that basis has a simple form. dx 83. (Matrices associated to linear maps) Let V;W be vector spaces of dimensions n;m respectively. Let T2 Hom(V;W) be a linear map from V to W. Show that there are ordered bases B=  n m v  V and C=fwg  W and an integer d minfn;mg such that the matrix A= j i i=1 j=1 (  1 i= j d a of T with respect to those bases satisfies a = , that is has the form i j i j 0 otherwise 0 1 1 B . C . . B C B C 1 B C B C 0 B C B C . . A . 0     1 2 4 (Hint1: study some examples, such as the matrices and ) (Hint2: start your 1 1 2 solution by choosing a basis for the image of T ). Extra credit: Finite fields 4. Let F be a field. (a) Define a mapi : (Z;+)(F;+) by mapping n2Z to the sum 1 ++ 1 n times. 0 F F Show that this is a group homomorphism. DEF If the mapi is injective we say that F is of characteristic zero. (b) Suppose there is a non-zero n2Z in the kernel ofi. Show that the smallest positive such number is a prime number p. DEF In that case we say that F is of characteristic p. (c) Show that in that casei induces an isomorphism between the finite fieldF =Z=pZ and p a subfield of F. In particular, there is a unique field of p elements up to isomorphism. 5. Let F be a field with finitely many elements. (a) Carefully endow F with the structure of a vector space overF for an appropriately chosen p p. r (b) Show that there exists an integer r 1 such that F has p elements. r RMK For every prime power q= p there is a fieldF with q elements, and two such fields q are isomorphic. They are usually called finite fields, but also Galois fields after their dis- coverer. Supplementary Problems I: A new field n o p p A. LetQ( 2) denote the set a+ b 2j a;b2Q R. p (a) Show thatQ( 2) is aQ-subspace ofR. p (b) Show thatQ( 2) is two-dimensional as aQ-vector space. In fact, identify a basis. p (c) Show thatQ( 2) is a field. p p (d) Let V be a vector space overQ( 2) and suppose that dim V = d. Show that Q( 2) dim V = 2d. Q 9Supplementary Problems II: How physicists define vectors Fix a field F. B. (The general linear group) (a) Let GL (F) denote the set of invertible n n matrices with coefficients in F. Show that n GL (F) forms a group with the operation of matrix multiplication. n (b) For a vector space V over F let GL(V) denote the set of invertible linear maps from V to itself. Show that GL(V) forms a group with the operation of composition. (c) Suppose that dim V = n Show that GL (F)' GL(V) (hint: show that each of the two F n n group is isomorphic to GL(F ). C. (Group actions) Let G be a group, X a set. An action of G on X is a map: G X X such that g(h x)=(gh) x and 1  x= x for all g;h2 G and x2 X (1 is the identity element of G G G). (a) Show that matrix-vector multiplication (g;v)7 gv defines an action of G= GL (F) on n n X = F . (b) Let V be an n-dimensional vector space over F, and letB be the set of ordered bases of V . n o n dimV n For g2 GL (F) and B=fvg 2B set gB= g v . Check that gB2B and å n i j i i i=1 j=1 j=1 that(g;B)7 gB is an action of GL (F) onB. n 0 0 (c) Show that the action is transitive: for any B;B2B there is g2 GL (F) such that gB= B . n (d) Show that the action is simply transitive: that the g from part (b) is unique. D. (From the physics department) Let V be an n-dimensional vector space, and letB be its set n n of bases. Given u2 V define a map f :B F by setting f (B)= a if B=fvg and u u i i=1 n u= a v . å i i i=1  (a) Show that af +f 0 =f 0. Conclude that the set f forms a vector space over u u au+u u u2V F. n (b) Show that the mapf :B F is equivariant for the actions of B(a),B(b), in that for each u  g2 GL (F), B2B, g f (B) =f (gB). n u u n (c) Physicists define a “covariant vector” to be an equivariant mapf :B F . LetF be the set of covariant vectors. Show that the map u7f defines an isomorphism VF. (Hint: u n n define a mapF V by fixing a basis B=fvg and mappingf7 a v iff(B)= a). å i i i=1 i=1 i n (d) Physicists define a “contravariant vector” to be a map f :B F such that f(gB) = t 1 t 1 n g (f(B)). Verify that (g;a)7 g a defines an action of GL (F) on F , that the set n 0 F of contravariant vectors is a vector space, and that it is naturally isomorphic to the dual 0 vector space V of V . Supplementary Problems III: Fun in positive characteristic E. Let F be a field of characteristic 2 (that is, 1 + 1 = 0 ). F F F 2 2 2 (a) Show that for all x;y2 F we have x+ x= 0 and(x+ y) = x + y . F (b) Considering F as a vector space overF as in 5(a), show that the map given by Frob(x)= 2 2 x is a linear map. 2 (c) Suppose that the map x7 x is actually F-linear and not onlyF -linear. Show that F=F . 2 2 RMK Compare your answer with practice problem 1. F. (This problem requires a bit of number theory) Now let F have characteristic p 0. Show that p the Frobenius endomorphism x7 x isF -linear. p 10CHAPTER 1 Constructions Fix a field F. 1.1. Direct sum, direct product 1.1.1. Simplest case (Lecture 2, 8/1/2014). CONSTRUCTION 13 (External direct sum). Let U;V be vector spaces. Their direct sum, de- noted UV , is the vector space whose underlying set is UV , with coordinate-wise addition and scalar multiplication. LEMMA 14. This really is a vector space. REMARK 15. The Lemma serves to review the definition of vector space. PROOF. Every property follows from the respective properties of U;V .  REMARK 16. More generally, can take the direct sum of groups. LEMMA 17. dim (UV)= dim U+ dim V . F F F REMARK 18. This Lemma serves to review the notion of basis. PROOF. Let B ;B be bases of U;V respectively. Thenf(u;0 )g tf(0 ;v)g is a U V V U u2B v2B U V basis of UV .  n m n+m EXAMPLE 19. R R 'R . (Lecture 3, 10/1/2014) A key situation is when U;V are subspaces of an “ambient” vector space W. LEMMA 20. Let W be a vector space, U;V W. Then Span (UV)=fu+ vj u2 U;v2 Vg. F PROOF. RHS contained in the span by definition. It is a subspace (non-empty, closed under addition and scalar multiplication) which contains U;V hence contains the span.  DEFINITION 21. The space in the previous lemma is called the sum of U;V and denoted U+V . LEMMA 22. Let U;V W. There is a unique homomorphism UV U+V which is the identity on U;V . PROOF. Define f((u;v))= u+ v. Check that this is a linear map.  PROPOSITION 23 (Dimension of sums). dim (U+V)= dim U+ dim V dim (U\V). F F F F PROOF. Consider the map f of Lemma 22. It is surjective by Lemma 20. Moreover Ker f = f(u;v)2 UVj u+ v= 0 g, that is W Ker f =f(w;w)j w2 U\Vg' U\V: Since dim Ker f+ dim Im f = dim(UV) the claim now follows from Lemma 17.  F F 11REMARK 24. This was a review of that formula. DEFINITION 25 (Internal direct sum). We say the sum is direct if f is an isomorphism. THEOREM 26. For subspaces U;V W TFAE (1) The sum U+V is direct and equals W; (2) U+V = W and U\V =f0g (3) Every vector w2 W can be uniquely written in the form w= u+ vv. PROOF. (1))(2): U+V = W by assumption, U\V = Ker f . (2))(3): the first assumption gives existence, the second uniqueness. (3))(1): existence says f is surjective, uniqueness says f is injective.  1.1.2. Finite direct sums (Lecture 4, 13/1/2013). Three possible notions: (UV)W, U (VW), vector space structure on UVW. These are all the same. Not just isomorphic (that is, not just same dimension), but also isomorphic when considering the extra structure of the copies of U;V;W. How do we express this? DEFINITION 27. W is the internal direct sum of its subspacesfVg if it spanned by them i i2I and each vector has a unique representation as a sum of elements of V (either as a finite sum of i non-zero vectors or as a zero-extended sum). REMARK 28. This generalizes the notion of “linear independence” from vectors to subspaces. LEMMA 29. Each of the three candidates contains an embedded copy of U;V;W and is the internal direct sum of the three images. PROOF. Easy.  PROPOSITION 30. Let A;B each be the internal direct sum of embedded copies of U;V;W. Then there is a unique isomorphism A B respecting this structure. PROOF. Construct.  REMARK 31. (1) Proof only used result of Lemma, not specific structure; but (2) proof im- plicitly relies on isomorphism to UVW; (3) We used the fact that a map can be defined using values on copies of U;V;W (4) Exactly same proof as the facts that a function on 3d space can be defined on bases, and that that all 3d spaces are isomorphic.  Dimension by induction. DEFINITION 32. Abstract arbitrary direct sum.  Block diagonality.  Block upper-triangularity. first structural result. 1.2. Quotients (Lecture 5, 15/1/2015) Recall that for a group G and a normal subgroup N, we can endow the quotient G=N with group structure(gN)(hN)=(gh)N.  This is well-defined, gives group.  Have quotient map q: G G=N given by g7 gN. 12 Homomorphism theorem: any f : G H factors as G G=Ker( f) follows by isomor- phism.  If N M G with both N;M normal then q(M)' M=N is normal in G=N and(G=N)=(M=N)' (G=M). Now do the same for vector spaces. LEMMA 33. Let V be a vector space, W a subspace. Let p : V V=W be the quotient as abelian groups. Then there is a unique vector space structure on V=W making p a surjective linear map. PROOF. We must set a(v+W)=av+W. This is well-defined and gives the isomorphism.  Properties persist. 1.3. Hom spaces and duality 1.3.1. Hom spaces (Lecture 5 continued). DEFINITION 34. Hom (U;V) will denote the space of F-linear maps U V . F U LEMMA 35. Hom (U;V) V is a subspace, hence a vector space. F 0 DEFINITION 36. V = Hom (V;F) is called the dual space. F 0 Motivation 1: in PDE. Want solutions in some function space V . Use that V is much bigger to 0 find solutions in V , then show they are represented by functions. 13Math 412: Problem Set 2 (due 22/1/2014) Practice f g P1 Let V be a family of vector spaces, and let A2 End(V)= Hom(V;V). i i i i i i2I L L (a) Show that there is a unique element A2 End( V) whose restriction to the image i i i2I i2I of V in the sum is A . i i L (b) Carefully show that the matrix of A in an appropriate basis is block-diagonal. i i2I P2 Construct a vector space W and three subspaces U;V ;V such that W = U V = U V 1 2 1 2 (internal direct sums) but V =6 V . 1 2 Direct sums 1. Give an example of V ;V ;V  W where V\V =f0g for every i=6 j yet the sum V +V +V 1 2 3 i j 1 2 3 is not direct. T r r r 2. LetfVg be subspaces of W with dim(V)(r 1)dimW. Show that V =6 f0g. å i i i i=1 i=1 i=1 3. (Diagonability) (a) Let T2 End(V). For eachl2 F let V = Ker(Tl). Let Spec (T)=fl2 Fj V 6=f0gg l F l be the set of eigenvalues of T . Show that the sum V is direct (the sum equals å l2Spec (T) l F V iff T is diagonable). (b) Show that a square matrix A2 M (F) is diagonable over F iff there exist n one-dimensional n L n n n subspaces V F such F = V and A(V) V for all i. i i i i i=1 Quotients 4. Let sl (F)=fA2 M (F)j TrA= 0g and let pgl (F)= M (F)=F I (matrices modulu scalar n n n n n matrices). Suppose that n is invertible in F (equivalently, that the characteristic of F does not divide n). Show that the quotient map M (F) pgl (F) restricts to an isomorphism n n sl (F)pgl (F). n n 5. Recall our axiom that every vector space has a basis. 1 (a) Show that every linearly independent set in a vector space is contained in a basis. (b) Let U W. Show that there exists another subspace V such that W = UV . (c) Let W = UV , and let p : W W=U be the quotient map. Show that the restriction of W to V is an isomorphism. Conclude that if UV ' UV then V ' V (c.f. problem 1 2 1 2 P2) 6. (Structure of quotients) Let V W with quotient mapp : W W=V . (a) Show that mapping U7p(U) gives a bijection between (1) the set of subspaces of W containing V and (2) the set of subspaces of W=V . (b) (The universal property) Let Z be another vector spaces. Show that f7 fp gives a linear bijection Hom(W=V;Z)fg2 Hom(W;Z)j V Kergg. 1 Directly, without using any form of transfinite induction 14n 7. For f :R R the Lipschitz constant of f is the (possibly infinite) number   j f(x) f(y)j def n k fk = sup j x;y2R ;x6= y : Lip jx yj n o n n Let Lip(R )= f :R Rjk fk ¥ be the space of Lipschitz functions. Lip n n PRA Show that f2 Lip(R ) iff there is C such thatj f(x) f(y)j Cjx yj for all x;y2R . n (a) Show that Lip(R ) is a vector space. n (b) Let1 be the constant function 1. Show thatk fk descends to a function on Lip(R )=R1. Lip n ¯ ¯ ¯ (c) For f2 Lip(R )=R1 show that f = 0 iff f = 0. Lip Supplement: Infinite direct sums and products CONSTRUCTION. LetfVg be a (possibly infinite) family of vector spaces. i i2I S (1) The direct product V is the vector space whose underlying space isf f : I Vj8i : f(i)2 Vg Õ i i i i2I i2I with the operations of pointwise addition and scalar multiplication.   L (2) The direct sum V is the subspace f2 Vj ij f(i)6= 0 ¥ of finitely Õ i i2I i i2i V i supported functions. A. (Tedium) (a) Show that the direct product is a vector space (b) Show that the direct sum is a subspace. (c) Letp : V V be the projection on the ith coordinate (p( f)= f(i)). Show that this i Õ i i i i2I ( is a surjective linear map. v j= i (d) Let s : V V be the map such that s(v)( j) = . Show that s is an Õ i i i2I i i i 0 j=6 i injective linear map. B. (Meat) Let Z be another vector space. L (a) Show that V is the internal direct sum of the imagess(V). i i i i2I (b) Suppose for each i2 I we are given f 2 Hom(V;Z). Show that there is a unique f2 i i L Hom( V) such that fs = f . i i i i2I (c) You are instead given g2 Hom(Z;V). Show that there is a unique g2 Hom(Z;Õ V) such i i i i thatp g= g for all i. i i 0 C. (What a universal property can do) Let S be a vector space equipped with maps s : V S, i i and suppose the property of 5(b) holds (for every choice of f 2 Hom(V;Z) there is a unique i i f2 Hom(S;Z) ...) 0 (a) Show that eachs is injective (hint: take Z= V , f the identity map, f = 0 if i=6 j). j j i i 0 (b) Show that the images of thes span S. i (c) Show that S is the internal direct sum of the S . i L (d) (There is only one direct sum) Show that there is a unique isomorphism j : S V i i2I 0 such that js =s (hint: construct j by assumption, and a reverse map using the ex- i i istence part of 5(b); to see that the composition is the identity use the uniqueness of the assumption and of 5(b), depending on the order of composition). 0 D. Now let P be a vector space equipped with mapsp : P V such that 5(c) holds. i i 0 (a) Show thatp are surjective. i 0 (b) Show that there is a unique isomorphismy : : P V such thatpy =p . Õ i2I i i i 15Supplement: universal properties E. A free abelian group is a pair (F;S) where F is an abelian group, S F, and (“universal property”) for any abelian group A and any (set) map f : S A there is a unique group homo- ¯ ¯ morphism f : G A such that f(s)= f(s) for any s2 S. The size S is called the rank of the free abelian group. (a) Show that(Z;f1g) is a free abelian group.   d d (b) Show that Z ;feg is a free abelian group. k k=1 0 0 0 (c) Let (F;S);(F ;S) be free abelian groups and let f : S S be a bijection. Show that f 0 ¯ extends to a unique isomorphism f : F F . (d) Let(F;S) be a free abelian group. Show that S generates F. (e) Show that every element of a free abelian group has infinite order. Supplement: Lipschitz functions DEFINITION. Let(X;d );(Y;d ) be metric spaces, and let f : X Y be a function. We say f X Y 0 is a Lipschitz function (or is “Lipschitz continuous”) if for some C and for all x;x2 X we have   0 0 d f(x); f(x)  Cd x;x : Y X Write Lip(X;Y) for the space of Lipschitz continuous functions, and for f2 Lip(X;Y) write n o 0 d ( f(x); f(x)) Y 0 k fk = sup j x6= x2 X for its Lipschitz constant. 0 Lip d (x;x) X F. (Analysis) (a) Show that Lipschitz functions are continuous. 1 n n (b) Let f2 C (R ;R). Show thatk fk = supfjÑ f(x)j : x2Rg. Lip (c) Show thatka f+bgk jajk fk +jbjkgk (“kk is a seminorm”). Lip Lip Lip Lip  n ¯ ¯ (d) Show that D f;g ¯ = f g ¯ defines a metric on Lip(R ;R)=R1. Lip (e) Generalize (a),(c),(d) to the case of Lip(X;R) where X is any metric space. (f) Show that Lip(X;R)=R1 is complete for all metric spaces X. 161.3.2. The dual space, finite dimensions (Lecture 6, 17/1/2014). CONSTRUCTION 37 (Dual basis). Let B=fbg  V be a basis. Write v2 V uniquely as i i2I v= a b (almost all a = 0) and setj(v)= a . å i i i i i2I i LEMMA 38. These are linear functionals. 0 PROOF. Representav+ v in the basis.  n EXAMPLE 39. V = F with standard basis, getj(x)= x . Note every functional has the form i i n j(x)= j(e)j(x). å i i=1 i REMARK 40. Alternative construction:j is the unique linear map to F satisfyingj(b )=d . i i i; j j LEMMA 41. The dual basis is linearly independent. If dim V ¥ it is spanning. F PROOF. Evaluate a linear combination at b .  j 0 REMARK 42. This isomorphism V V is not canonical: the functional j depends on the i whole basis B and not only on b , and the dual basis transforms differently from the original basis i under change-of-basis. Also, the proof used evaluation – let’s investigate that more. 0 PROPOSITION 43 (Double dual). Given v2V consider the evaluation map e : V F given by v 00 e (j)=j(v). Then v7 e is a linear injection V,V , an isomorphism iff V is finite-dimensional. v v 0 V PROOF. The vector space structure on V (and on F in general) is such that e is linear. That v 0 the map v7 e is linear follows from the linearity of the elements of V . For injectivity let v2 V v be non-zero. Extending v to a basis, let j be the element of the dual such that j (v)= 1. Then v v 0 00 e (j )6= 0 so e =6 0. If dim V = n then dim V = n and thus dim V = n and we have an v v v F F F isomorphism.  00 The map V , V is natural: the image e of v is intrinsic and does not depend on a choice of v basis. 1.3.3. The dual space, infinite dimensions (Lecture 7, 20/1/2014). Interaction with past con- 0 0 structions: (V=U) V (PS2), 0 0 0 LEMMA 44. (UV)' UV . PROOF. Universal property.  0 0 n n COROLLARY 45. Since(F)' F, it follows by induction that(F )' F . What about infinite sums? L ‘ 0 0  The universal property gives a bijection( V) V . i i2I i2I i ‘  Any Cartesian product W has a natural vector space structure, coming from point- i i2I wise addition and scalar multiplication: – Note that the underlying set is ¡ W = f fj f is a function with domain I and8i2 I : f(i)2 Wg i i i2I ( ) = f : I Wj f(i)2 W : i i i2I 17 RMK: AC means Cartesian products nonempty, but our sets have a distin- guished element so this is not an issue. def 0 0 – Definea(w) +(w) = (aw + w) . This gives a vector space structure. i i i2I i i i2I i2I – Denote the resulting vector space W and called it the direct product of the W . Õ i i i2I L 0 0  The bijection( V) V is now a linear isomorphism in fact, the vector space Õ i i2I i2I i structure on the right is the one transported by the isomorphism. We now investigateÕ W in general. i i  Note that it contains a copy of each W (map w2 W to the sequence which has w in the i i ith position, and 0 at every other position).  And these copies are linearly independent: if a sum of such vectors from distinct W is i zero, then every coordinate was zero. L  Thus W contains W as an internal direct sum. Õ i i i i2I – This subspace is exactly the subsetfw2 Wj supp(w) is finiteg. Õ i i L – And in fact, that subspace proves that W exists. i i2I – ButÕ W contains many other vectors – it is much bigger. i i N N EXAMPLE 46. R R . COROLLARY 47. The dual of an infinite-dimensional space is much bigger than the sum of the duals, and the double dual is bigger yet. 1.3.4. Question: so we only have finite sums in linear algebra. What about infinite sums? ¥ Answer: no infinite sums in algebra. Definition of a = A from real analysis relies on analytic å n n=1 properties of A (close to partial sums), not algebraic properties. But, calculating sums can be understood in terms of linear functionals. N LEMMA 48 (Results from Calc II, reinterpreted). Let SR denote the set of sequences a ¥ such that a converges. å n n=1 N N (1)R  SR is a linear subspace. ¥ (2) S: SR given byS(a)= a is a linear functionals. å n n=1 Philosophy: Calc I,II made element-by-element statements, but using linear algebra we can ex- press them as statements on the whole space. Now questions about summing are questions about intelligently extending the linear functional S to a bigger subspace. BUT: if an extension is to satisfy every property of summing series, it is actually the trivial (no) extension. For more information let’s talk about limits of sequences instead (once we can generalize limits just apply that to partial sums of a series). ¥ N DEFINITION 49. Let c` R be the sets of convergent, respectively bounded sequences. ¥ LEMMA 50. c` are subspaces, and lim : cR is a linear functional. n¥ 1 N N N EXAMPLE 51. Let C :R R be the Cesàro map(Ca) = a . This is clearly linear. å n N N n=1 1 0 Let CS= C (c) be the set of sequences which are Cesàro-convergent, and set L2 CS by L(a)= lim (Ca). This is clearly linear (composition of linear maps). For example, the sequence n¥ 1 (0;1;0;1;) now has the limit . 2 18LEMMA 52. If a2 c then Ca2 c and they have the same limit. Thus L above is an extension of lim . n¥ 0 ¥ THEOREM 53. There are two functionals LIM;lim 2(` ) (“Banach limit”, “limit along w ultrafilter”, respectively) such that: (1) They are positive (map non-negative sequences to non-negative sequences); (2) Agree with lim on c; n¥ (3) And, in addition ¥ ¥ (a) LIM S= LIM where S: ` ` is the shift. (b) lim (a b )=(lim a )(lim b ). w n n w n w n 1.3.5. The invariant pairing (Lecture 8, 22/1/2014). 0  Pairing VV 0  Bilinear forms in general, equivalence to maps V U . n m  Pairing F  F via matrix; matrix representation of general bilinear form (“Gram ma- trix”)  Non-degeneracy: Duality = non-degen-pairing = isom to dual 0 0  Identification of duals using pairings: Riesz representation theorems for C(X) ,H . 1.3.6. The dual of a linear map (Lecture 9, 24/1/2014). 0 0 0 0 CONSTRUCTION 54. Let T2 Hom(U;V). Set T 2 Hom(V ;U ) by(T j)(v)=j(T v). 0 0 LEMMA 55. This is a linear map Hom(U;V) Hom(V ;U ). An isomorphism of U;V finite- dimensional. 0 0 0 LEMMA 56. (T S) = S T 19Math 412: Problem Set 3 (due 29/1/2014) Practice 0 1 0 1 0 1 0 1 1 1 1 0 3 A A A A P1 Let u = 1 , u = 1 , u = 1 , u= 1 as vectors inR . 1 2 3 1 1 1 0  0 3 (a) Construct an explicit linear functionalj2 R vanishing on u ;u . 1 2 3 (b) Show thatfu ;u ;ug is a basis onR and find its dual basis. 1 2 3 (c) Evaluate the dual basis at u. m 0 P2 Let V be n-dimensional and letfjg 2 V . i i=1 (a) Show that if m n there is a non-zero v2 V such thatj(v)= 0 for all i. Interpret this as i a statement about linear equations. m (b) When is it true that for each x2 F there is v2 V such that for all i,j(v)= x ? i i Banach limits ¥ N Recall that` R denote the set of bounded sequences (the sequences a such that for some N N N M we havejaj M for all i). Let S:R R be the shift map(Sa) = a . A subspace UR i n+1 n is shift-invariant if S(U) U. If U is shift-invariant a function F with domain U is called shift- N invariant if FS= F (example: the subset cR of convergent sequences is shift-invariant, as is the functional lim: cR assigning to every sequence its limit). P3 (Useful facts) ¥ N (a) Show that` is a subspace ofR . N N ¥ ¥ (b) Show that S:R R is linear and that S(` )=` . N f g (c) Let UR be a shift-invariant subspace. Show that the set U = Sa aj a2 U is a 0 subspace of U. N (d) In the case U =R of sequences of finite support, show that U = U. 0 (e) Let Z be an auxiliary vector space. Show that F2 Hom(U;Z) is shift-invariant iff F vanishes on U . 0 ¥ ¥ 1. Let W =fSa aj a2` g` . Let1 be the sequences everywhere equal to 1. ¥ ¥ (a) Show that the sum W+R1` is direct and construct an S-invariant functionalj : ` R such thatj(1)= 1. ¥ (b) (Strengthening) For a2` setkak = sup jaj. Show that if a2 W and x2 R then n ¥ n ka+ x1k jxj. (Hint: consider the average of the first N entries of the vector a+ x1). ¥ 0 ¥ SUPP Letj2(` ) be shift-invariant, positive, and satisfyj(1)= 1. Show that liminf a  n¥ n j(a) limsup a and conclude that the restriction ofj to c is the usual limit. n n¥ n 1+(1) 0 ¥ 2. (“choose one”) Letj2(` ) satisfyj(1)= 1. Let a be the sequence a = . n 2 1 (a) Suppose thatj is shift-invariant. Show thatj(a)= . 2 (b) Suppose that j respects pointwise multiplication (if z = x y then j(z)= j(x)j(y)). n n n Show thatj(a)2f0;1g. 20

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