1 Sequences. 2 Series. 2 SERIES Analysis Study Guide

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1 2 SERIES Anlysis Study Guide 1 Sequences Def: An ordered field is field F nd totl order < (for ll x, y, z F ): (i) x < y, y < x or x = y, (ii) x < y, y < z x < z (iii) x < y x + z < y + z (iv) 0 < y, x 0 < xy. 1 Def: The Archimeden property on n ordered field F is x, y F, x, y > 0, there exists N N such tht n x > y. 2 Fct: b = c d r+sc rb+sd = b for ll, b, c, d, r, s Z with rb + sd 0.3 Def: A rel number L is the limit of sequence of rel numbers ( n ) n=1 if for every ε > 0, there is n N N such tht n L < ε for ll n N. Then, ( n ) converges to L. 4 Thm: Squeeze Theorem Suppose three sequences ( n ), (b n ), (c n ) stisfy n b n c n nd lim n n = lim n c n = L. Then lim n b n = L. 5 Prop: If ( n ) n=1 converges, then the set { n n N } is bounded. 6 Thm: If lim n n = L, lim n b n = M, nd α R, then (i) lim n n + b n = L + M. (ii) lim n α n = αl. (iii) lim n n b n = LM. (iv) lim n n bn = L M if M Thm: For sequence n 0, we hve lim n n = + if n only if lim n n = 0. 8 Thm: Lest Upper Bound Principle Every nonempty subset S of R tht is bounded bove hs supremum. Similrly, every nonempty subset S of R tht is bounded below hs n infimum. 9 Thm: Monotone Convergence Theorem A monotone incresing sequence tht is bounded bove converges. A monotone decresing sequence tht is bounded below converges. 10 Thm: Let ( n ) be sequence. (i) If lim n is defined, then lim inf n = lim sup n = lim n. (ii) if lim inf n = lim sup n = L, then lim n = L. 11 Def: A subsequence ( n ) n=1 is new sequence ( nk ) k=1 = ( n 1, n2,... ) where n 1 < n 2 <. 12 Thm: If the sequence ( n ) converges, then every subsequence converges to the sme limit. 13 Thm: Every sequence ( n ) hs monotonic subsequence. 14 Cor: Let ( n ) be sequence. There exists monotonic subsequence whose limit is lim sup n nd there exists monotonic subsequence shows limit is lim inf n. 15 Def: Let ( n ) be sequence in R. A subsequentil limit is ny rel number (or symbol +, ) tht is the limit of some subsequence ( nk ). 16 Thm: Let ( n ) be ny sequence in R, nd let S denote the set of subsequentil limits of ( n ). (i) S is nonempty. (ii) sup S = lim sup n nd inf S = lim inf n. (iii) lim n exists if n only if S hs single element, nmely lim n. 17 Thm: Let S denote the set os subsequentil limits of sequence ( n ). Suppose (b n ) is sequence in S R nd tht t = lim t n. Then t S. 18 Lm: Nested Intervls Lemm Suppose tht I n = [ n, b n ] = { x R n x b n } re nonempty closed intervls such tht I n+1 I n for ech n 1. Then the intersection n 1 I n is nonempty. 19 Thm: Bolzno-Weierstrss Theorem Every bounded sequence of rel numbers hs convergent subsequence. 20 Fct: For bounded sequence ( n ), lim sup n (lim inf n ) is the lrgest (smllest) possible vlue for convergent subsequence. 21 Def: A sequence ( n ) n=1 is clled Cuchy sequence if for every ε > 0, there is n integer N such tht m n < ε for ll m, n N. 22 Def: A subset S of R is sid to be complete if every Cuchy sequence converges to point in S. 23 Thm: Completeness Theorem A sequence of rel numbers converges if nd only if it is Cuchy sequence. In prticulr, R is complete Series Def: Given sequence ( n ) n=1, the infinite series n=1 n = lim n n k=1 k converges if the limit exists, diverges otherwise. 25 Fct: Some series cn be solved using telescoping sum, by cncelling elements between sequence terms. 26 Thm: nth term test If lim n 0, then n diverges. 27

2 3 TOPOLOGY OF R N Anlysis Study Guide Thm: Cuchy Criterion for Series The following re equivlent for series n=1 n. (i) The series converges. (ii) For every ε > 0, there is n N N so tht k=n+1 k < ε for ll n N. (iii) For every ε > 0, there is n N N so tht m k=n+1 k < ε if n, m N. 28 Prop: If k 0 for k 1 nd s n = n k=1 k, then either (i) (s n ) n=1 is bounded bove, in which cse n=1 n converges. (ii) (s n ) n=1 is unbounded, in which cse n=1 n diverges. 29 Def: A sequence ( n ) n=0 is geometric series with rtio r if n+1 = r n for ll n 0, or equivlently n = 0 r n. Thm: Convergence of Geometric Series A geometric series n=0 rn converges to 1 r if r < 1.30 Def: A sequence ( n ) n=0 is p-series with power p if n = n for some, p R. p Thm: Convergence of p-series A p-series n=1 n converges if nd only if p > p Thm: Comprison Test Consider two sequences ( n ), (b n ) with n b n for ll n 1. If (b n ) is summble, then ( n ) is summble nd n=1 n n=1 b n. If ( n ) is not summble, then (b n ) is not summble. 32 Thm: Rtio Test A series n of nonzero terms (i) Converges bsolutely if lim sup n+1 / n < 1, (ii) Diverges if lim inf n+1 / n > Thm: Root Test Suppose tht n 0 for ll n nd let l = lim sup n n. If l < 1, then n=1 n converges bsolutely, nd if l > 1, the series diverges. 34 Thm: Integrl Test If f : R R is positive nd decresing, then n=a f(n) converges if nd only if lim b A f(x)dx exists (nd is finite). 35 Thm: Limit Comprison Test If n=1 b n is convergent series of positive numbers b n nd lim sup n b n < then n=1 n converges. 36 Def: A sequence is lternting if it hs the form ( 1) n n or ( 1) n+1 n where n 0 for ll n Lm: For x > 1, m N, (1 + x) m > 1 + mx. 38 Thm: Leibniz Alternting Series Test Suppose tht ( n ) n=1 is monotone decresing sequence nd tht lim n n = 0. Then the lternting series n=1 ( 1)n n converges. 39 Cor: Suppose tht ( n ) n=1 is monotone decresing positive sequence nd tht lim n n = 0. Then the difference between the sum of the lternting series n=1 ( 1)n n nd the Nth prtil sum is t most N. Def: A series n=1 n is clled bsolutely convergent if the series n=1 n converges. A series tht converges but is not bsolutely convergent is clled conditionlly convergent. 40 Def: A rerrngement of series n=1 n is nother series with the sme terms in different order. This cn be described by permuttion π of the nturl numbers N determining the series n=1 π(n). 41 Thm: For n bsolutely convergent series, every rerrngement converges to the sme limit. Lm: Let n=1 n be convergent series. Denote the positive terms s b 1, b 2,... nd the other terms s c 1, c 2,.... (i) If n=1 n is bsolutely convergent, then so re both n=1 b n nd n=1 c n, nd n=1 n = n=1 b n n=1 c n. (ii) If n=1 n is conditionlly convergent, then n=1 b n nd n=1 c n both diverge. 42 Thm: Rerrngement Theorem If n=1 is conditionlly convergent series, then for every rel number L, there is rerrngement tht converges to L. 43 Lm: Summtion by Prts Lemm Suppose (x n ) nd (y n ) re sequences of rel numbers. Define X n = n k=1 x k nd Y n = n k=1 y n. Then m n=1 x ny n + m n=1 X ny n+1 = X m Y m Thm: Dirichlet s Test Suppose tht ( n ) n=1 is sequence of rel numbers with bounded prtil sums. If (b n ) n=1 is sequence of postive numbers decresing monotoniclly to 0, then the series n=1 nb n converges. 45 Thm: Abel s Test Suppose tht n=1 n converges nd (b n ) is monotonic convergent sequence. Then, n=1 nb n converges Topology of R n Def: The dot product or inner product of two vectors x, y R n is x, y = n=1 x iy i. 47 Thm: Schwrz Inequlity For ll x, y R n, x, y x y. Equlity holds if nd only if x nd y re coliner. 48 Thm: Tringle Inequlity The tringle inequlity holds for the Eucliden norm on R n : x + y x + y for ll x, y R n. Moreover, equlity holds if nd only if either x = 0 or y = cx with c 0. 49

3 4 FUNCTIONS Anlysis Study Guide Def: A set V = {x 1,..., x m } R n is orthonorml if v i, v j = 0 when i j nd v i, v i = 1. If m = 0, then V spns R n nd is clled n orthonorml bsis. 50 Lm: Let {v 1,..., v n } be n orthonorml set in R n. Then m i=1 ix i = ( m i=1 i 2) 1/2. 51 Def: A sequence of points (x k ) in R n converges to point if for every ε > 0, there is n integer N so tht x k < ε for ll k > N. In this cse, write lim n x k =. 52 Lm: Let (x k ) be sequence in R n. Then lim k x k = if nd only if lim k x k = Lm: A sequence x k = (x k,1,..., x k,n ) in R n converges to point = ( 1,..., n ) if nd only if ech coordinte converges: lim k x k,i = i for 1 i n. 54 Def: A sequence x k in R n is Cuchy if for every ε > 0, there is n integer N so tht x k x l < ε for ll k, l > N. A set S R n is complete if every Cuchy sequence of points in S converges to point in S. 55 Thm: Completeness Theorem for R n Every Cuchy sequence in R n converges. Thus, R n is complete. 56 Def: A point x is limit point of subset A R n if there is sequence ( k ) k=1 with k A such tht x = lim k k. A set A R n is closed if it contins ll of its limit points. 57 Def: A point x is cluster point of subset A R n if there is sequence ( n ) n=1 with n A \ {x} such tht x = lim n n. 58 Prop: If A, B R n re closed, then A B is closed. If { A i i I } is fmily of closed subsets of R n, then i I A i is closed. 59 E.g.: Let A n = B n 1 n (0). The fmily of sets {A n } n N hs every set closed, but n N A n = B 1 (0), which is not closed. 60 Def: If A is subset of R n, the closure of A is the set A consisting of ll limit points of A. 61 Def: The bll bout in R n of rdius r is the set B r () = { x R n x < r }. A subset U R n is open if for every U, there is some r > 0 so tht the bll B r () is contined in U. 62 Prop: Let A R n. Then A is the smllest closed set contining A. In prticulr, A = A. 63 Thm: Dulity of Open nd Closed Sets A set A R n is open if nd only if the complement of A, A = { x R n x / A } is closed. 64 HERE MARKS THE END OF EXAM 1 MATERIAL Prop: If U nd V re open subsets of R n then U V is n open subset of R n. If { U i i I } s fmily of open subsets of R n, then i I U i is open. 65 E.g.: Let A n = B n+1 (0). The fmily of sets {A n } n N hs every set open, but n N A n = B 1 (0), which is closed. 66 n Def: The interior, int X, of set X is the lrgest open set contined in X. If int X =, then X hs empty interior. 67 Def: A subset A R n is compct if every sequence ( k ) k=1 of points in A hs convergent subsequence ( k i ) i=1 with limit = lim i ki in A. 68 Def: A subset S of R n is clled bounded provided tht there is rel number R such tht S is contined in the bll B R (0). 69 Lm: A compct subset of R n is closed nd bounded. 70 Lm: If C is closed subset of compct subset of R n, then C is compct. 71 Lm: The cube [, b] n is compct subset of R n. 72 Thm: Heine-Borel Theorem A subset of R n is compct if nd only if it is closed nd bounded. 73 Thm: Cntor s Intersection Theorem If A 1 A 2 A 3 is decresing sequence of nonempty compct subsets of R n, then k 1 A k is not empty. 74 Def: A set whose closure hs no interior is nowhere dense. A point x of set A is isolted if there is n ε > 0 such tht the bll B ε (x) intersects A only in the singleton {x}. A set A is perfect if ech point x A is the limit of some sequence in A \ {x} Functions Def: Let S R n nd let f : S R m. If S is cluster point (limit point of S \ {}) then point v R n is limit of f t if for every ε > 0 there is n r > 0 so tht f(x) v < ε whenever 0 < x < r nd x S. Write lim x f(x) = v. 76 Def: Let S R n nd let f : S R m. f is continuous t S if for every ε > 0, there is n r > 0 such tht, for ll x S with x < r, we hve f(x) f() < ε. Moreover, f is continuous on S if it is continuous t ech point S. If f is not continuous t, f is discontinuous t. 77

4 4 FUNCTIONS Anlysis Study Guide Def: A function f : S R m is clled Lipschitz if there is constnt C such tht f(x) f() C x for ll x, S. The Lipschitz constnt of f is the smllest choice of C for this condition to hold. 78 Def: A function f : [, b] R stisfies Lipschitz condition of order α > 0 if there is some positive constnt M so tht f(x 1 ) f(x 2 ) M x 1 x 2 α. 79 Prop: Every Lipschitz function is continuous. 80 Cor: Every liner trnsformtion A from R n to R m is Lipschitz, nd therefore continuous. 81 Def: The coordinte functions re, π j : R n R is given s π j (x 1,..., x n ) = x j for 1 j n, nd ɛ i (t) = te i. 82 Def: A function f : R n R m hs removble singulrity t if lim x f(x) exists, but does not equl f(). 83 Def: The limit of f s x pproches from the right is L if for every ε > 0, there is n δ > 0 so tht f(x) L < ε for ll < x < + δ, written lim x + f(x) = L. Similrly, define the limit from the left, lim x f(x) = L. 84 Def: When function f on R hs limits from the left nd right tht re different, we sy f hs jump discontinuity. A function on n intervl is clled piecewise continuous if on every finite subintervl, it hs only finite number of jump discontinuities nd is continuous t ll other ponts. 85 E.g.: The Heviside function H(x) is defined to be 0 for ll x < 0 nd 1 for x E.g.: The ceiling function x hs countbly mny jump discontinuities. 87 E.g.: For ny subset A R n, the chrcteristic function of A is χ A (x), which equls 1 if x A, nd 0 otherwise. 88 E.g.: Let f(x) = 0 when x / Q nd then f(x) = 1 q when x = p q, in reduced form. Then, f(x) is continuous only t irrtionl points, nd discontinuous on the rtionls. 89 Def: Sy tht the limit of function f(x) hs x pproches is + if for every positive integer N, there is n r > 0 so tht f(x) > N for ll 0 < x < r. We write lim x f(x) = +. We define the limit lim x f(x) = similrly. 90 Def: A function f is symptotic to the curve g if lim x f(x) g(x) = Def: A subset V S R n is open in S (or reltively open) if there is n open set U in R n such tht U S = V. 92 Thm: For function f mpping S R n into R m, the following re equivlent: (i) f is continuous on S. (ii) Sequentil chrcteriztion of continuity For every convergent sequence (x n ) n=1 with lim n x n = S. lim n f(x n ) = f(). (iii) Topologicl chrcteriztion of continuity For every open set U in R m, the set f 1 (U) = { x S f(x) U} is open in S. 93 Thm: If f, g re functions from common domin S into R m, S such tht lim x f() = u nd lim x g(x) = v, then (i) lim x f(x) + g(x) = u + v. (ii) lim x αf(x) = αu. When the rnge is contined in R, sy lim x f(x) = u nd lim x g(x) = v, then (iii) lim x f(x)g(x) = uv, nd (iv) lim x f(x)/g(x) = u/v provided v Thm: If f, g re functions form S to R m tht re continuous t S nd α R, then (i) f + g is continuous t. (ii) αf is continuous t. When the rnge is contined in R, (iii) fg is continuous t (iv) f/g is continuous t provided tht g() Def: A function f is rtionl function if f(x) = p(x)/q(x) where p, q R[x] nd q 0. Rtionl functions re continuous except where q(x) = Def: If function f : S T nd g : T R m, then the composition of g nd f, denoted g f is the function tht sends x to g(f(x)). 97 Thm: Suppose f : S T nd g : T R m. If f is continuous t S nd g is continuous t f() T, then g f is continuous t. 98 Thm: Let C be compct subset of R n, nd let f be continuous function from C into R m. Then the imge set f(c) is compct. 99 Thm: Extreme Vlue Theorem Let C be compct subset of R n nd let f be continuous function from C into R. Then there re points nd b in C ttining the minimum nd mximum vlues of f on C. Tht is, f() f(x) f(b) for ll x C. 100

5 5 INTRO TO CALCULUS Anlysis Study Guide Def: A function f : R R m is periodic function if there exists d > 0 such tht f(x) = f(x + d) for ll x R. The lest such d is clled the period of f. Then, f is d-periodic. 101 Def: A function f : S R m is uniformly continuous if for every ε > 0 there is δ > 0 so tht f(x) f() < ε whenever x < δ for x, S. 102 Prop: Every Lipschitz function is uniformly continuous. 103 Cor: Every liner trnsformtion from R n to R m is uniformly continuous. 104 Cor: Let f be differentible rel-vlued function on [, b] with bounded derivtive; tht is, there is M > 0 so tht f (x) M for ll x b. Then f is uniformly continuous on [, b]. 105 Thm: Suppose tht C R n is conpct nd f : C R n is continuous. Then f is uniformly continuous on C. 106 Thm: Intermedite Vlue Theorem If f is continuous rel-vlued function on [, b] with f() < 0 < f(b), then there exists point c (, b) such tht f(c) = Cor: If f is continuous rel-vlued function on [, b], then f([, b]) is closed intervl. 108 Def: A pth in S R n from to b, both poionts in S, is the imge of continuous function γ from [0, 1] into S such tht γ(0) = nd γ(1) = b. 109 Cor: Suppose tht S R n nd f is continuous rel-vlued function on S. If there is pth from to b in S nd f() < 0 < f(b), then there is point c on the pth so tht f(c) = Def: A function f is incresing on n intervl (, b) if f(x) f(y) whenever < x y < b. It is strictly incresing on (, b) if f(x) < f(y) whenever < x < y < b. Similrly, define decresing nd strictly decresing functions. All of these functions re clled monotone. 111 Prop: If f is n incresing function on the inervl (, b), then the one-sided limits of f exist t ech point c (, b) nd lim x c f(x) = L f(x) lim x c + f(x) = M. For decresing functions, the inequlities re reversed. 112 Cor: The only type of discontinuity tht monotone function on n intervl cn hve is jump discontinuity. 113 Cor: If f is monotone function on [, b] nd the rnge of f intersects every nonempty open intervl in [f(), f(b)] then f is continuous. 114 Thm: A monotone function on [, b] hs t most countbly mny discontinuities. 115 Thm: Let f be continuous strictly incresing function on [, b]. Then f mps [, b] one-to-one nd onto [f(), f(b)]. Moreover the inverse function f 1 is lso continuous nd strictly incresing. 116 Prop: Let f : S R m, S R n be continuous function. If T S is compct, f(t ) is compct. 117 E.g.: The Cntor function c : [0, 1] [0, 1] is n onto function defined such tht c is constnt over the intervls not in the Cntor set, but is strictly incresing over the Cntor set. Also, c(c) = [0, 1] Intro to Clculus f(x Def: A funciton f : (, b) R is differentible t point x 0 (, b) if lim 0+h) f(x 0) h 0 h exists. We write f (x 0 ) for this limit. 119 Def: When f is differentible t x 0, we define the tngent line to f t x 0 to be the liner function T (x) = f(x 0 ) + f (x 0 )(x x 0 ). 120 Prop: If f is differentible t x 0, then it is continuous t x 0. Differentible functions re continuous. 121 Lm: Let f be function on [, b] tht is differentible t x 0. Let T (x) be the tngent line to f t x 0. Then T is the f(x) T (x) x x 0 = unique liner function with the property lim x x0 Cor: If f(x) is function on (, b) nd x 0 (, b), then the following re equivlent: (i) f is differentible t x 0. (ii) There is liner function T (x) nd function ε(x) on (, b) such tht lim x x0 ε(x) = 0 nd f(x) = T (x) + ε(x)(x x 0 ). (iii) There is function ϕ(x) on (, b) such tht f(x) = f(x 0 ) + ϕ(x)(x x 0 ) nd lim x x0 ϕ(x) exists. 123 Thm: Arithmetic of Derivtives Let f nd g be differentible functions t the point. Ech of the functions cf (c constnt), f + g, fg, nd f/g re differentible t, except f/g if g() = 0. The formuls re 124 (i) (cf) () = c f () (ii) (f + g) () = f () + g () (iii) Product rule (fg) () = f()g () + f ()g() (iv) Quotient rule (f/g) () = g()f () f()g () g 2 () if g() 0. Thm: The Chin Rule Suppose tht f is defined on [, b] nd hs rnge contined in [c, d]. Let g be defined on [c, d]. Suppose tht f is differentible t x 0 [, b] nd g is differentible t f(x 0 ). Then the composition h(x) = g(f(x)) is defined, nd h (x 0 ) = g (f(x 0 ))f (x 0 ). 125

6 6 INTEGRATION Anlysis Study Guide E.g.: The clss of functions f(x) = x sin(1/x) for x > 0 nd f(0) = 0 for > 0 is differentible on [0, ), but the derivtive function is not continuous t Def: A function f(x) is even if f( x) = f(x) nd odd when f( x) = f(x). 127 HERE MARKS THE END OF EXAM 2 MATERIAL Thm: Fermt s Thoerem Let f be continuous function on n intervl [, b] tht tkes its mximum or minimum vlue t point x 0. Then, exctly one of the following holds: (i) x 0 is n endpoint or b, (ii) f is not differentible t x 0, (iii) f is differentible t x 0 nd f(x 0 ) = Thm: Rolle s Theorem Suppose tht f is function tht is continuous on [, b] nd differentible on (, b) such tht f() = f(b) = 0. THen there is point c (, b) such tht f (c) = Thm: Men Vlue Theorem Suppose tht f is function tht is continuous on [, b] nd differentible on (, b). Then there is point c (, b) such tht f (C) = f(b) f() b. 130 Cor: Let f be differentible function on [, b]. 131 (i) If f (x) is (strictly) positive, then f is (strictly) incresing. (ii) If f (x) is (strictly) negtive, then f is (strictly) decresing. 132 (iii) If f (x) = 0 t every x (, b), then f(x) is constnt. 133 (iv) If g is differentible on [, b] with g (x) = f (x), then there is constnt c such tht f(x) = g(x) + c. 134 Def: If twice differentible function f hs f (x) positive on n intervl [, b] then f is convex or concve up. If f (x) is negtive, then f is concve or concve down. The points where f (x) chnges sign re clled inflection points. 135 Thm: Drboux s Theorem/Intermedite Vlue Theorem for Derivtives If f is differentible on [, b] nd f () < L < f (b), then there is point x 0 in (, b) t which f (x 0 ) = L. 136 Thm: Let f is one-to-one continuous function on n open intervl I, nd let J = f(i). If f is differentible t x 0 I nd if f (x 0 ) 0, then f 1 is differentible t y 0 = f(x 0 ) nd (f 1 ) (y 0 ) = 1 f (x 0) Integrtion Def: Let f : [, b] R be bounded function. 138 (i) A prtition of [, b] is finite set P = { = x 0 < x 1 < < x n 1 < x n = b}. Set j = x j x j 1 nd define the mesh of prtition P s mesh(p ) = mx 1 j n j. 139 (ii) Let the mximum nd minimum be M j (f, P ) = sup{ f(x) x j 1 x x j } nd m j (f, P ) = inf{ f(x) x j 1 x x j }. (iii) Let the upper (Drboux) sum nd lower (Drboux) sum be U(f, P ) = n j=1 M j(f, P ) j nd L(f, P ) = n j=1 m j(f, P ) j. (iv) If given n evlution sequence X = {x j 1 j n} with x j [x j 1, x j ] the Riemnn sum is I(f, P, X) = n j=1 f(x j ) j. 140 (v) A prtition R is refinement of prition P provided P R. If P nd Q re prtitions, then R is common refinement if P Q R. Lm: If R is refinement of P, then L(f, P ) L(f, R) U(f, R) U(f, P ). 141 Cor: If P nd Q re ny two prtitions of [, b], L(f, P ) U(f, Q). 142 Def: Define the lower Drboux integrl, L(f) = sup P L(f, P ) nd the upper drboux integrl, U(f) = inf P U(f, P ). Note tht L(f) U(f). 143 A bounded function f on finite intervl [, b] is clled Riemnn integrble if L(f) = U(f). In this cse, we write L(f) = f(x)dx = U(f).144 Thm: Riemnn s Condition Let f(x) : [, b] R be bounded function. The following re equivlent: (i) f is Riemnn integrble. (ii) For ech ε > 0, there is prtition P so tht U(f, P ) L(f, P ) < ε. 145 Cor: Let f be bounded rel-vlued function on [, b]. If there is sequence of prtitions of [, b], P n so tht lim n U(f, P n ) L(f, P n ) = 0, then f is Riemnn integrble. Moreover, if X n is ny choice of points x n,j selected from ech intervl of P n, then lim n I(f, P n, X n ) = f(x)dx. Thm: Let f(x) : [, b] R be bounded function. The following re equivlent: 146 (i) f is Riemnn integrble. (ii) For ech ε > 0, there is prtition P so tht U(f, P ) L(f, P ) < ε. (iii) Cuchy Criterion for Integrbility For every ε > 0, there is δ > 0 so tht every prtition Q such tht mesh(q) < δ stisfies U(f, Q) L(f, Q) < ε. 147

7 7 RIEMANN-STIELTJES INTEGRATIONAnlysis Study Guide Thm: Let f(x) : [, b] R be bounded function, then f is Riemnn integrble with f(x)dx = L if nd only if for every ε > 0, there is δ > 0 so tht every prtition Q such tht mesh(q) < δ nd every choice of evlution sequence X stisfies I(f, Q, X) L < ε. 148 Thm: Every monotone function on [, b] is Riemnn integrble. 149 Thm: Every continuous function on [, b] is integrble. 150 Def: Sy tht f is Riemnn integrble on [, ) if the improper integrl f(x)dx := lim b f(x)dx exists.151 Thm: Arithmetic of Integrls Let f nd g be integrble function on [, b] nd c R. Then (i) cf is integrble nd cf(x)dx = c f(x)dx. (ii) f + g is integrble nd (f + g)(x)dx = f(x)dx + b g(x)dx.152 Thm: If f nd g re integrble on [, b] nd if f(x) g(x) for ll x [, b], then f(x)dx g(x)dx.153 Thm: If f is integrble on [, b], then f is integrble on [, b] nd f(x)dx f(x) dx.154 Thm: If f : [, b] R with c (, b) nd f is integrble on [, c] nd [c, b] then f is integrble on [, b] with f(x) = c f(x)dx + c f(x)dx.155 Thm: Fundmentl Theorem of Clculus I Let f be bounded Riemnn integrble function on [, b] nd let F : [, b] R be defined s F (x) = x f(t)dt. Then F is continuous, nd if f is continuous t point x 0, then F is differentible t x 0 with F (x 0 ) = f(x). 156 Def: A function f on [, b] hs n ntiderivtive if there is continuous function F : [, b] R such tht F (x) = f(x) for ll x [, b]. 157 Cor: Fundmentl Theorem of Clculus II Let f be continuous function on [, b]. Then f hs n ntiderivtive. Moreover, if G is ny ntiderivtive of f, then f(x)dx = G(b) G().158 Lm: Suppose tht f : [, b] R is n integrble function bounded by M : [, b] R. Then f(t)dt M(b ).159 Thm: Integrtion by Prts GIven two differentible functions f : [, b] R nd g : [, b] R, the integrl f (x)g(x)dx = f(x)g(x) b f(x)g (x)dx. 160 Thm: Substitution/Chnge of Vrible Let u be C 1 function on [, b] nd let f be continuous on [c, d] contining the rnge of u. Then f(u(x))u (x)dx = u(b) u() f(t)dt.161 Thm: Intermedite/Men Vlue Theorem for Integrls If f is continuous function on [, b], then there is point c (, b) such tht 1 b f(x)dx = f(c) Riemnn-Stieltjes Integrtion Def: Consider bounded functions f, g : [, b] R. Given prtition of [, b], P = { = x 0 < x 1 < < x n = b}, nd n evlution sequence X = (x 1, x 2,, x n) for P, we define the Riemnn-Stieltjes sum, or briefly, the R-S sum, for f w.r.t. g using P nd X, s I g (f, P, X) = n f(x j)[g(x j ) g(x j 1 )]. j=1 We sy f is Riemnn-Stieltjes integrble with respect to g, denoted f R(g), if there is number L so tht for ll ε > 0, there is prtition P ε so tht, for ll prtitions P with P P ε nd ll evlution sequences X for P, we hve I g (f, P, X) L < ε. Thm: If f 1 R(g) nd f 2 R(g) on [, b], nd c 1, c 2 R, then c 1 f 1 + c 2 f 2 R(g) on [, b] nd c 1 f 1 + c 2 f 2 dg = c 1 f 1 dg + c 2 f 2 dg. If f R(g 1 ) nd f R(g 2 ) on [, b], then f R(g 1 + g 2 ) on [, b] nd fd(g 1 + g 2 ) = fdg 1 + fdg 2.

8 7 RIEMANN-STIELTJES INTEGRATIONAnlysis Study Guide Finlly, if < b < c nd f is R-S integrble w.r.t. g on both [, b] nd [b, c], then it is R-S integrble w.r.t. g on [, c] nd c fdg = fdg + Thm: Suppose f, g : [, b] R re bounded nd ϕ : [c, d] [, b] is strictly incresing, continuous function onto [, b]. Let F = f ϕ nd G = g ϕ. If f R(g) on [, b] then F R(G) on [c, d] nd d c F dg = Thm: Let f, g : [, b] R be bounded functions. If f R(g) on [, b], then g R(f) on [, b] nd fdg + c b fdg. fdg. gdf = g(b)f(b) g()f(). Thm: If f : [, b] R hs f R(g), where g : [, b] R is C 1, then fg is Riemnn integrble on [, b] nd fdg = f(x)g (x)dx. Def: Let f nd g be bounded functions on [, b]. As usul, let P = {x 0 < x 1 < < x n } be prtition of [, b] nd recll tht M j (f, P ) = sup f(x) nd m j (f, P ) = inf f(x). x j 1 x x j x j 1 x x j Define the upper sum with respect to g nd the lower sum with respect to g to be U g (f, P ) = L g (f, P ) = n M i (f, P )[g(x i ) g(x i 1 )] i=1 n m i (f, P )[g(x i ) g(x i 1 )]. i=1 Rmrk: If g is incresing then, for ll evlution sequences X, L g (f, P ) I g (f, P, X) U g (f, P ). Lm: (Refinement Lemm) Let f, g be bounded functions on [, b] nd P, R prtitions of [, b]. incresing. If R is refinement of P, then Assume tht g is L g (f, P ) L g (f, R) U g (f, R) U g (f, P ). Cor: Let f, g be bounded functions on [, b] nd ssume tht g is incresing. If P nd Q re ny two prtitions of [, b], L g (f, P ) U g (f, Q). Def: We define L g (f) = sup P L g (f, P ) nd U g (f) = inf P U g (f, P ). Prop: (Riemnn-Stieltjes Condition) Let f, g be bounded functions on [, b] nd ssume tht g is incresing on [, b]. TFAE: 1. f is R-S integrble w.r.t. g. 2. U g (f) = L g (f), nd 3. for ech ε > 0, there is prtition P so tht U g (f, P ) L g (f, P ) < ε. Def: Given function f : [, b] R nd prtition [, b], sy P = { = x 0 < x 1 < < x n = b}, then the vrition of f over P is n V (f, P ) := f(x i ) f(x i 1 ). i=1

9 8 NORMED VECTOR SPACES Anlysis Study Guide Rmrk: If P is prtition of [, b] nd Q is prtition of [b, c], then for function f : [, c] R, V (f, P Q) = V (f, P ) + V (f, Q). Rmrk: If R is refinement of P, then V (f, P ) V (f, R). Def: The (totl) vrition of f on [, b] is V b f = sup {V (f, P ) : P prtition of [, b]}. We sy f is of bounded vrition on [, b] if V b f is finite. Lm: If < b < c nd f : [, c] R is given, then V c f = V b f + V c b f. Thm: If f : [, b] R is of bounded vrition on [, b], then there re incresing functions g, h : [, b] R so tht f = g h. Thm: If f : [, b] R is bounded by M, g : [, b] R is of bounded vrition nd f R(g) on [, b], then fdg M V b g. Thm: If f : [, b] R is continuous nd g : [, b] R is incresing, then f R(g) on [, b]. Cor: If f : [, b] R is continuous nd g : [, b] R is of bounded vrition, then f R(g) on [, b]. 8 Normed Vector Spces Def: Let V be vector spce over R. A norm on V is function on V tking vlues in [0, + ) with the following properties: 1. (positive definite) x = 0 if nd only if x = 0, 2. (homogeneous) αx = α x for ll x V nd α R, nd 3. (tringle inequlity) x + y x + y for ll x, y V. E.g.: Let V = R n. Then the following re norms: ( n ) 1/2 x = (x 1,, x n ) 2 = x i 2 x 1 = (x 1,, x n ) 1 = i=1 n x i i=1 x = (x 1,, x n ) = mx 1 i n x i. E.g.: Fix rel number p in [1, ). Define the L p [, b] norm on C[, b] by ( 1/p b f p = f(x) dx) p. Def: A complete normed vector spce V is clled Bnch spce. Prop: A sequence x n in normed vector spce V converges to vector x if nd only if for ech open set U contining x, there is n integer N so tht x n U for ll n N. Def: A subset K of normed vector spce V is compct if every sequence (x n ) of points in K hs subsequence (x ni ) which converges to point in K.

10 8 NORMED VECTOR SPACES Anlysis Study Guide 8.1 Inner Product Spces 8.1 Inner Product Spces Def: An inner product on vector spce V is function x, y on pirs (x, y) of vectors in V V tking vlues in R stisfying the following properties: 1. (positive definiteness) x, x 0 for ll x V nd x, x = 0 only if x = (symmetry) x, y = y, x for ll x, y V. 3. (bilinerity) For ll x, y, z V nd sclrs α, β R, αx + βy, z = α x, z + β y, z. Given n inner product spce, it is esy to check tht the following definition gives us norm: E.g.: The spce C[, b] cn be given n inner product x = x, x 1/2. f, g = f(x)g(x)dx, which gives rise to the L 2 norm. Thm: (Cuchy-Schwrz Inequlity) For ll vectors x, y in n inner product spce V, x, y x y. Equlity holds if nd only if x nd y re coliner. Cor: Let V be n inner product spce with induced norm. Then the inner product is continuous (i.e., if x n converges to x nd y n converges to y, then x n, y n converges to x, y ). 8.2 Orthonorml Sets Def: Two vectors x nd y re clled orthogonl if x, y = 0. A collection of vectors {e n : n S} in V is clled orthonorml if e n = 1 for ll n S nd e n, e m = 0 for n m S. This set is clled n orthonorml bsis if in ddition this set is mximl with respect to being n orthonorml set. Prop: An orthonorml set is linerly independent. An orthonorml bsis for finite-dimensionl inner product spce is bsis. Lm: The functions { 1, 2 cos nθ, 2 sin nθ : n 1 } form n orthonorml set in C[ π, π] with the inner product f, g = 1 π 2π π f(θ)g(θ)dθ. Def: A trigonometric polynomil is finite sum f(θ) = A 0 + N A k cos kθ + B k sin kθ. Def: Denote the Fourier series of f C[ π, π] by f = A 0 + A n cos nθ + B n sin nθ, where A 0 = 1 π 2π f(t)dt, nd π n=1 for n 1, A n = 1 π π π f(t) cos ntdt nd B n = 1 π π f(t) sin ntdt. π Lm: Let {e 1,, e n } be n orthonorml set in n inner product spce V. If M is the subspce spnned by {e 1,, e n }, then every vector x M cn be written uniquely s n α i e i, where α i = x, e i. In other words, the set {e 1,, e n } i=1 is linerly independent. Moreover, for ech y v with y, e i = β i, nd ech x = n α j e j in M, x, y = n α i β i. Cor: If V is n inner product spce of finite dimension n, then it hs n orthonorml bsis {e i : 1 i n} nd the inner product is given by n n n α i e i, β j e j = α i β i. i=1 j=1 k=1 i=1 j=1 i=1

11 8 NORMED VECTOR SPACES Anlysis Study Guide 8.3 Orthogonl Expnsions in IPS 8.3 Orthogonl Expnsions in IPS Def: A projection is liner mp P such tht P 2 = P. In ddition, sy tht P is n orthogonl projection if ker P is orthogonl to Rn P. Rmrk: Observe tht ker P = Rn (I P ) becuse P x = 0 if nd only if (I P )x = x. Further, x = (I P )y implies P x = (P P 2 )y = 0. Thus, when P is n orthogonl projection, the vectors P x nd (I P )x re orthogonl. Thm: (Projection Theorem) Let A = {e 1,, e n } be n orthonorml set in n IPS V nd let M be the subspce spnned by A. Define P : V M by P y = n y, e j e j, for ech y V. Then P is the orthogonl projection onto M nd Moreover, for ll v M, j=1 y n y, e j 2. j=1 y v 2 = y P y 2 + P y v 2. In prticulr, P y is the closest vector in M to y. Prop: (Bessel s Inequlity) Let {e n : n S} be n orthonorml set in n inner product spce V. For ech vector x V, x, e n 2 x 2. n S Def: A complete inner product spce is clled Hilbert spce. ( 1/2 E.g.: The spce l 2 consists of ll sequences x = (x n ) n=1 such tht x 2 = xn) 2 is finite. The inner product on n=1 l 2 is given by x, y = x n y n. n=1 Thm: The spce l 2 is complete. Proof. Forthcoming. Thm: (Prsevl s Theorem) Let S N nd E = {e n : n S} be n orthonorml set in Hilbert spce H. Then the subspce M = spn {E} consists of ll vectors x = α n e n, where the coefficient sequence (α n ) n=1 belongs to l 2. Further, if x is vector in H, then x M if nd only if x, e n 2 = x 2. n S n S Cor: Let E = {e n : n S} be n orthonorml set in Hilbert spce H. Then there is continuous liner orthogonl projection P E of H onto M = spn {E} given by P E (x) = x, e n e n. Cor: If E = {e i : i 1} is n orthonorml bsis for Hilbert spce H, every vector x H my be uniquely expressed s x = α i e i, where α i = x, e i. i=1 8.4 Finite-Dimensionl Normed Spces Lm: If {v 1,, v n } is LI set in normed vector spce (V, ), then there exist positive constnts 0 < c < C such tht for ll = ( 1,, n ) R n, we hve c 2 n i v i C 2. i=1 Rmrk: This effectively mens tht every finite-dimensionl normed vector spce hs the sme topology s (R n, 2 ) in the sense tht they hve the sme convergent sequences, the sme open/closed sets, etc. Cor: A subset of finite-dimensionl normed vector spce is compct if nd only if it is closed nd bounded. Cor: A finite-dimensionl subspce of normed vector spce is complete, nd in prticulr it is closed. Thm: Let (V, ) be normed vector spce, nd let W be finite-dimensionl subspce of V. Then for ny v V, there is t lest one closest point w W so tht v w = inf { v w : w W }. n S

12 9 LIMITS OF FUNCTIONS Anlysis Study Guide 9 Limits of Functions 9.1 Limits of Functions Def: Let (f n ) be sequence of functions from S R n into R m. This sequence converges pointwise to function f if lim n f n(x) = f(x) for ll x S. Def: Let (f n ) be sequence of functions from S R n to R m. This sequence converges uniformly to function f if for every ε > 0, there is n integer N so tht f n (x) f(x) < ε for ll s S nd n N. Thm: For sequence of functions (f n ) in C b (S, R m ) (the spce of ll bounded continuous functions from S to R m ), (f n ) converges uniformly to f if nd only if lim n f n f = 0. E.g.: The sequence f n (x) = x n for x [0, 1] converges pointwise to χ {1}. The functions f n re polynomils, nd hence not only continuous but even smooth; while the limit function hs discontinuity t the point 1. For ech n 1, we hve f n (1) = 1 nd so f n χ {1} = sup x n 0 = 1. 0 x<1 So f n does not converge in the uniform norm. Indeed, to contrdict the definition, tke ε = 1/2. For ech n, let x n = 2 1/2n. Then f n (x n ) χ {1} (x n ) = 1 > ε. 2 Hence no integer N stisfies the definition. 9.2 Uniform Convergence nd Continuity Thm: Let (f n ) be sequence of continuous functions mpping subset of S of R n into R m tht converges uniformly to function f. Then f is continuous. Thm: (Completeness of C(K)) If K is compct set, the spce C(K) of ll continuous functions on K with the sup norm is complete. 9.3 Uniform Convergence nd Integrtion Thm: (Integrl Convergence Theorem) Let (f n ) be sequence of continuous functions on the closed intervl [, b] converging uniformly to f(x) nd fix c [, b]. Then the functions F n (x) = x c f n (t)dt for n 1 converge uniformly on [, b] to the function F (x) = x c f(t)dt. Cor: Suppose tht (f n ) is sequence of continuously differentible functions on [, b] such tht (f n) converges uniformly to function g nd there is point c [, b] so tht lim f n(c) = γ exists. Then (f n ) converges uniformly to n differentible function f with f(c) = γ nd f = g. Prop: Let f(x, t) be continuous function on [, b] [c, d]. Define F (x) = d f(x, t)dt. Then F is continuous on [, b. Thm: (Leibniz s Rule) Suppose tht f(x, t) nd d x c f(x, t)dt is differentible nd F (x) = c f(x, t) re continuous functions on [, b] [c, d]. Then F (x) = d c f(x, t)dt. x

13 9 LIMITS OF FUNCTIONS Anlysis Study Guide 9.4 Series of Functions 9.4 Series of Functions Thm: Let (f n ) be sequence of continuous functions from subset S of R n into R m. If f n (x) converges uniformly, then it is continuous. Def: Let S R n. We sy tht series of functions f k from S to R m is uniformly Cuchy on S if for every ε > 0 there is N so tht l f i (x) ε whenever x S nd l > k N. i=k+1 Thm: A series of functions converges uniformly if nd only if it is uniformly Cuchy. Thm: (Weierstrss M-Test) Suppose tht n (x) is sequence of functions on S R k into R m nd (M n ) is sequence of rel numbers so tht n = sup n (x) M n for ll x S. x S If M n converges, then the series n (x) converges uniformly on S. n 1 n=1 E.g.: The function f(x) = 2 k cos(10 k πx) is continuous on R nd differentible t no point in R. k 1 n=1 9.5 Power Series Def: A power series is series of functions of the form n x n = x + 2 x 2 +. n=0 Thm: (Hdmrd s Theorem) Given power series n x n, there is R [0, + ) {+ } so tht the series converges n=0 for ll x with x < R nd diverges for ll x with x > R. Moreover, the series converges uniformly on ech closed intervl [, b] contined in ( R, R). Finlly, if α = lim sup n 1/n, then n + if α = 0 R = 0 if α = + if α (0, + ). 1 α We cll R the rdius of convergence of the power series. Thm: (Term-by-Term Differentition of Series) If f(x) = n x n hs rdius of convergence R > 0, then n n x n 1 hs rdius of convergence R, f is differentible on ( R, R) nd, for x ( R, R), Further, n=0 f (x) = n=0 n n x n 1. n=1 n n + 1 xn+1 hs rdius of convergence R nd, for x ( R, R), x 0 f(t)dt = 9.6 Compctness nd Subsets of C(K) n=0 n n + 1 xn+1. Def: A fmily of functions F mpping set S R n into R m is equicontinuous t point S if for every ε > 0, there is n r > 0 such tht f(x) f() < ε whenever x < r nd f F. n=1

14 10 METRIC SPACES Anlysis Study Guide The fmily F is equicontinuous on set S if it is equicontinuous t every point in S. The fmily F is uniformly equicontinuous on S if for ech ε > 0, there is n r > 0 such tht f(x) f(y) < ε whenever x y < r, x, y S nd f F. Lm: Let K be compct subset of R n. A compct subset F of C(K, R m ) is equicontinuous. Prop: If F is n equicontinuous fmily of functions on compct set, then it is uniformly equicontinuous. Def: A subset S of K is clled n ε-net of K if K S B ε (). A set K is totlly bounded if it hs finite ε-net for every ε > 0. Lm: Let K be bounded subset of R m. Then K is totlly bounded, mn. Cor: Let K be compct subset of R m. Then K contins sequence {x i : i 1} tht is dense in K. Moreover, for ny ε > 0, there is n integer N so tht {x i : 1 i N} forms n ε-net for K. Thm: (Arzel-Ascoli) Let K be compct subset of R n. A subset F of C(K, R m ) is compct if nd only if it is closed, bounded, nd equicontinuous. 10 Metric Spces 10.1 Definitions nd Exmples Def: Let X be set. A metric on set X on set X is function d : X X [0, ) with the following properties: (1) d(x, y) = 0 if nd only if x = y, (2) d(x, y) = d(y, x) for ll x, y X, (3) d(x, z) d(x, y) + d(y, z) for ll x, y, z X. A metric spce is set X with metric d, denoted (X, d). E.g.: If X is subset of normed spce V, define d(x, y) = x y. E.g.: Define metric on Z by ρ 2 (n, n) = 0 nd ρ 2 (m, n) = 2 d, where d is the lrgest power of 2 dividing m n. E.g.: If X is closed subset of R n, let K(X) denote the collection of ll nonempty compct subsets of X. If A is compct subset of X nd x X, we define dist(x, A) = inf x. Then we define the Husdorff metric on A K(X) by { d H (A, B) = mx { = mx sup A sup } dist(, B), sup dist(b, A) b B } inf b, sup inf b. A b B b B A Prop: When X R n is closed, then (K(X), d H ) is complete metric spce. Thm: Let f : (X, ρ) (Y, σ). TFAE: 1. f is continuous on X; 2. for every convergent sequence (x n ) n=1 with lim x n = in X, we hve lim f(x n ) = f(); nd 3. f 1 (U) is open for every open set U Y. Thm: The spce C b (X) of ll bounded continuous functions on X with the sup norm f = sup { f(x) : x X} is complete. Rmrk: Given metric spce (X, d), (X, d) (the bounded metric spce) hs the sme topology s (X, d).

15 11 DISCRETE DYNAMICAL SYSTEMSAnlysis Study Guide 10.2 Compct Metric Spces 10.2 Compct Metric Spces Thm: (Borel-Lebesgue) For metric spce X, TFAE: 1. X is compct. 2. Every collection of closed subsets of X with the FIP hs nonempty intersection. 3. X is sequentilly compct. 4. X is complete nd totlly bounded. Prop: A metric spce X is complete if nd only if every decresing sequence of nonempty closed sets (blls) with rdii going to zero hs nonempty intersection. 11 Discrete Dynmicl Systems 11.1 Fixed Points nd the Contrction Principle Def: For metric spce (X, d), cll T : X X contrction if there is c (0, 1) such tht d(t x, T y) cd(x, y) for ll x, y X. Tht is, T stisfies Lipschitz condition with c < 1. For T : X X cll x 0 X fixed point for T if T x 0 = x 0. E.g.: Let X = R, T (x) = mx + b (n ffine mp). If m = 0, b is fixed point nd imt = B. If b = 0 then T (x) = mx nd x = 0 is fixed point. If m 1, this is the only fixed point. If m < 1, then for ny x T, T n (x) x. If m > 1, then T n (x) diverges. If m = 1, then T (T (x)) = T ( x) = x so for every x 0, we hve period 2 orbit. ( For generl b, we cn see inductively tht x n = m n x + b(1 + m + + m n 1 ) = m n x + b b 1 m 1 m n 1 m ) for m 1. If m < 1, then this converges to. If m > 1 nd x 0, this sequence diverges to. To find fixed points, we hve x = T (x) = mx + b implies x = b/(1 m) for m 1. If m > 1 nd x n = T (x n 1 ), then x n 1 = T 1 (x n ). Now T 1 is n ffine mp with slope 1/m < 1. Define x n = (T 1 ) n (x), which converges to b/(1 m) s n. It follows tht T n (x) diverges for ll x b/(1 m). It turns out tht T is contrction whenever m < 1. Notice we hve fixed points even for m 1. Thm: (Bnch Contrction Principle) Let (X, d) be complete metric spce. Let T : X X be contrction. Then T hs unique fixed point x, nd for ll x X, T n (x) x. Moreover for ny x X, 11.2 Iterted Function Systems d(x, T n (x)) c n d(x, x) cn d(x, T (x)). 1 c Def: We cll liner mp T : R n R n similitude if there is r > 0 such tht for ll x, y R n, T x T y = r x y. If r < 1, then T is contrction, clled contrctive similitude. An iterted function system (IFS) is finite set of contrctive similitudes. Fct: If A 1,, A r, B 1,, B r K(X), then d H (A 1 A r, B 1 B r ) mx {d H (A 1, B 1 ),, d H (A r, B r )}. Thm: Let X R n be closed. Suppose {T 1,, T r } is n IFS on X where T i hs Lipschitz constnt s i. Define T on K(X) by T (A) = T 1 (A) T r (A), nd let s = mx {s 1,, s r }. Then 1. T is contrction on K(X) with Lipschitz constnt s, 2. There is unique A K(X) such tht A = T (A), nd 3. For ny B K(X), d H (T k (B), A) s k d H (B, A) s k /(1 s)d H (T (B), B). E.g.: Concretely, how do we find the mgicl A for given IFS {T 1,, T r }? Consider words w in {1,, r} nd let w denote the length of w. Given word i 1 i 2 i m = w, define mp T w : X X by T i1 T i2 T im. Notice tht T w is contrctive similitude with constnt s i1 s im s w (where the s ij re the constnts for T ij respectively nd s is their mx). There re three pproches: (1) Find B K(X) such tht T (B) B. Then A = k=1 T i (B).

16 12 APPROXIMATION BY POLYNOMIALS Anlysis Study Guide (2) For ech word w, let w be the fixed point of T w. Then A = { w : w word in 1,, r}. (3) Pick ny point A. Then A = {T w () : w word in 1,, r}. Thm: (Fixed points of IFS s re dense in K(R n )) For ny C R n compct nd ε > 0, there is n IFS {T 1,, T r } so tht its fixed point set A stisfies d H (A, C) < ε. 12 Approximtion by Polynomils 12.1 Tylor Series Def: For f : [, b] R, n times differentible t c (, b), the n th Tylor polynomil for f t c is P n (x) = f(c) + f (c)(x c) + f (x 2)2 (c) + + f (n) (x c)n (c). 2! n! Lm: P n is the unique polynomil of degree t most n such tht P n (k) (c) = f (k) (c). Thm: Suppose f is n + 1 times differentible on [, b]. Let x, c (, b). There is d between x nd c such tht f(x) P n (x) = f (n+1) (d) Furthermore, if f (n+1) is bounded by M on [, b], then f(x) P n (x) M (x c)n+1. (n + 1)! x c n+1. (n + 1)! Def: Formlly, we sy f is O(g) ner R if there is some M > 0, δ > 0 such tht f(x) M g(x) for ll x with 0 < x δ. Prop: Rules for computtion: O(f) ± O(g) = O(mx {f, g}), O(f)O(g) = O(fg). E.g.: O((x ) m ) ± O((x ) n ) = O((x ) min{m,n} ), nd O((x ) m )((x ) n ) = O((x ) m+n ). Rmrk: Tylor s Theorem sys O(f P n ) = O((x c) n+1 ) How not to Approximte Function Thm: (Weierstrss Approximtion Theorem) Let f be ny continuous rel-vlued function on [, b]. Then there is sequence of polynomils p n tht converges uniformly to f on [, b] Bernstein s Proof of the Weierstrss Theorem Def: We define the Bernstein polynomils to be P n k (x) = ( n k) x k (1 x) n k. We define polynomil B n f by (B n f)(x) = n f k=0 ( ) k Pk n (x) = n n f k=0 ( k n ) ( n k Prop: The mp B n is liner nd monotone. Tht is, for ll f, g C[0, 1] nd α R, (1) B n (f + g) = B n f + B n g. (2) B n (αf) = αb n f (3) B n f 0 if f 0 (4) B n f B n g if f g (5) B n f B n g if f g. ) x k (1 x) n k.

17 NOTES Anlysis Study Guide NOTES Notes 1 Donsig, 825 Notes, 08/29. 2 Dongig, 825 Notes, 08/29. 3 Donsig, 825 Notes, 08/31. 4 D&D Definition p D&D. Theorem p D&D. Proposition p. 42; Ross, Theorem 9.1. p D&D. Theorem p. 42; Ross, Thm pp Ross. Theorem p D&D. Theorem p D&D. Theorem p. 47; Ross. Thm p Ross. Theorem p Ross. Definition p Ross. Theorem p Ross. Theorem p Ross. Corollry p Ross. Definition p Ross. Theorem p Ross. Theorem p D&D. Lemm p D&D. Theorem p Donsig, 825 Notes, 09/ D&D. Definition p. 56; Ross. Definition p D&D. Definition p D&D. Theorem p. 56; Ross. Theorem p D&D, Def p D&D, Exmple p Donsig, 825 Notes, 09/ D&D Theorem p. 69; Ross, Definition 14.3, Theorem pp D&D, Prop , p D&D, Theorem , p H,W&T, Exmple 3, p D&D, Theorem p. 71; Ross, Theorem p Ross, Theorem p D&D, Theorem , p.72.; Ross, Theorem p Donsig, 825 Notes, 09/ Donsig, 825 Notes, 09/ D&D, Definition p Donsig, 825 Notes, 09/26. D&D. p D&D. Theorem p D&D. Definition p D&D. Definition p D&D. Lemm p D&D, Theorem p D&D. Lemm p D&D. Theorem p D&D. Problem 3.4.G. Solution in 825 Problem Set D&D. p D&D. Theorem p D&D. Theorem p D&D. p D&D. Lemm p D&D. Definition p D&D. Lemm p D&D. Lemm p D&D. Definition p D&D. Theorem p D&D. Definition p D&D. 4.4.N. p D&D. Proposition p D.S. 10/ D&D. Definition p D&D. Definition p D&D. Proposition p D&D. Theorem p D&D. Proposition p D.S. 10/ D&D. p D&D. Definition p D&D. p D&D. Lemm p D&D. Lemm p D&D. Lemm p D&D. Theorem p D&D. Theorem p D&D. p D&D. Def p D&D. Def p D&D. Def p D&D. 5.5.K. p D&D. Prop p D&D. Cor p D&D. p D&D. p D&D. Def p D&D. Def p D&D. Ex p D&D. p D&D. Ex p D&D. Ex p D&D. Def p D&D. p D&D. p D&D. Thm p D&D. Thm p D&D. Thm p D&D. p D&D. p D&D. p D&D. Thm p D&D. Thm p D&D. 5.4.H. p D&D. Def p D&D. Prop p D&D. Cor p d&d. Cor p D&D. Thm p D&D. Thm p D&D. Cor p D&D. p D&D. Cor p D&D. Def p D&D. Prop p D&D. Cor p D&D. Cor p D&D. Thm p D&D. Thm p Not in the book or notes. 118 D&D. Exmple pp D&D Def p.141; Ross Def 28.1 pp D&D p D&D. Prop p.142; Ross Thm 28.2 p D&D Lemm p D&D Cor p Ross Thm 28.3 p D&D Thm p.145; Ross Thm 28.4 p D&D Exmple p D&D Exercise 6.1.E p D&D Thm p.148; Ross Thm 29.1 p D&D Thm p.148; Ross Thm 29.2 p D&D Thm p.149; Ross Thm 29.3 p D&D Cor p Ross Cor 29.7 p Ross Cor 29.4 p Ross Cor 29.5 p.216

18 13 COUNTEREXAMPLES Anlysis Study Guide 135 D&D p D&D 6.2.I p.152; Ross Thm 29.8 p Ross Thm 29.9 pp D&D Def p.154; Ross Def 32.1 pp Ross Def 32.6 p Ross Def 32.8 p D&D Lm p.155; Ross Lm 32.2 p D&D Cor p.155; Ross Lm 32.3 p Ross Thm 32.4 p D&D Def p.156; Ross Def 32.1 p D&D Thm p.156; Ross Thm 32.5 p D&D Thm p Ross Thm 32.7 p D&D Thm 6.3.8, (4) p D&D Thm p.160; Ross Thm 33.1 p D&D Thm p.160; Ross Thm 33.2 p D&D 6.3.S p Ross Thm 33.3 p Ross Thm 33.4 p Ross Thm 33.5 p Ross Thm 33.6 p D&D Thm p.165; Ross Thm 34.1 p D&D p D&D Cor p.165; Ross Thm 34.3 p D&D Lm p D&D p.167; Ross Thm 34.2 p D&D p.167; Ross Thm 34.4 p D&D 6.4.C p.168; Ross Thm 33.9 p Counterexmples