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# INTRODUCTION TO ANALYSIS I MATH 321

Rice University

GPA 3.86

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This 8 page Class Notes was uploaded by Jayde Lang on Monday October 19, 2015. The Class Notes belongs to MATH 321 at Rice University taught by Stephen Semmes in Fall. Since its upload, it has received 13 views. For similar materials see /class/224928/math-321-rice-university in Mathematics (M) at Rice University.

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Date Created: 10/19/15

Math 321 Calculus Handout 2 Let 17 b be real numbers with a lt b7 and let f be a real valued function on the open interval 17 b If a S p lt b7 then the limit of f as x a p with p lt z lt b may be denoted hm f96 or fp 1 ram when it exists Similarly7 if a lt p 3 b7 then the limit of f as x a p with a lt z lt p may be denoted 113 we or NH lt2 when it exists It a lt p lt b7 then 13 fx 3 exists if and only if the one sided limits exist are equal Suppose now that f 17 a R is also monotone increasing In this case7 the one sided limits exist at each point in 177 with fp inff96 i 10 lt 96 lt b 4 and fp7 supfxaltxltp 5 ln particular7 fp S f10 S fp 6 If f is bounded from below on 177 then fa exists and is equal to the in mum of f on 177 and if f is bounded from above on 177 then fb7 exists and is equal to the supremum of f on a7b Note that fp S W S fq7 7 whena plttltq b Math 321 Handout 28 Let M7dac7y7 N7pu7v be metric spaces A mapping f M a N is said to be bounded if fM is a bounded set in N Let CbM7 N be the space of bounded continuous mappings from M into N The supremum metric on CbM7 N is de ned by f17f2 suppf17 f2 00 E M for f17 f2 E CbM7 N It is easy to see that this is a metric on CbM7 N7 and that a sequence f j il of elements of CbM7 N converges to f E CbM7 N in the supremum metric if and only if fib l converges to f uniformly Suppose that N is complete7 and let us show that CbM7 N is complete with respect to the supremum metric Let fib l be a Cauchy sequence in CbM7N7 so that for every 6 gt 0 there is a positive integer L such that 1902 f1 lt 6 when j7l Z L ln particular7 fjx 1 is a Cauchy sequence in N for every 90 E M7 which converges to an element fx of N since N is complete One can check that pfjx7 S 6 for every ac E M when j Z L 7 which means that fj 1 converges uniformly to f and that f is continuous7 as desired Math 321 Handout 10a Let Mdxy be a metric space We say that E Q M is countably compact if for every open covering U aeA of E in M there is a set A1 Q A with only nitely or countably many elements such that E Q UaeAl Ua If E is compact then there is an A1 Q A with only nitely many elements which has this property Countable sets are automatically countably compact and more generally countable unions of countably compact sets are countably compact Suppose that M is countably compact and let 6 gt 0 be given The family of open balls Bxc 90 E M is an open covering of M and so countable compactness implies that there is an E6 Q M with only nitely or countably many elements such that M E UerS Bx ie EE is c dense in M Thus M is separable Conversely suppose that M is separable and let 8 be a base for the topology of M with only nitely or countable many elements Let U aeA be any open covering of M and let A be the set of V E B such that V Q UcK for some 04 E A For each 04 E A UcK is equal to the union of the V E B such that V Q Ua because 8 is a base for the topology of M Hence UVeAV M For each V E A let aV be an element of A such that V Q UMV and let A1 be the set of aV V E A It follows that A1 has only nitely or countable many elements since A does and that M UaeA1 UH Thus M is countably compact If M is totally bounded then M is separable and therefore countably compact The limit point property also implies that every covering by a sequence of open sets can be reduced to a nite subcovering from which we may conclude that M is compact Math 321 Calculus Handout 5 Let a b be real numbers with a lt b and let f be a continuous real valued function on the closed interval ab The classical Riemann integral fltzgt dz lt1 can be de ned in various ways as a limit of nite Riemann sums A key point is that f is uniformly continuous on ab which ensures that different approximations to the integral converge to the same value For example one can show that the sequence of Riemann sums associated to a sequence of partitions of ab with maximal step size tending to 0 is a Cauchy sequence and therefore converges One can also show that the limit of the sequence of Riemann sums is independent of the particular sequence of partitions Alternatively one can consider the upper and lower Riemann sums associated to any partition of ab Uniform continuity implies that the supremum of the lower sums is equal to the in mum of the upper sums and the integral of f can be de ned as their common value It is easy to extend the Riemann integral to functions with nitely many jump discontinuities One can go further and de ne the Riemann integrabil ity of a real valued function f on ab by the property that the supremum of the lower sums of f over all partitions of ab is equal to the in mum of the corresponding upper sums A famous theorem states that a bounded function on 1 b is Riemann integrable if and only if it is continuous almost everywhere and the Lebesgue integral extends the Riemann integral to a much broader class of functions By construction the integral of f is linear in f If a lt r lt b then fltzgtdz 17m dz rm dz lt2 Math 321 Handout 27a Let E be a set and let us consider complex valued functions on E If a pair of sequences fj1 of functions on E converge uniformly to functions f fon E respectively then it is easy to see that fj converges uniformly to f f on E If fjbil converges uniformly to f on E and C is a complex number then 0 f 1 converges uniformly to Cf on E If a pair of uniformly bounded sequences of functions on E converge uniformly then the corresponding sequence of products converges uniformly to the product of limits lf ajbil is a sequence of functions on E and 1311 is a sequence of nonnegative real numbers such that lajxl g Aj for every j Z 1 and 90 E E and 221 A converges then 2791 1790 converges absolutely for every 90 E E by the comparison test and Weierstrass made the nice observation that the partial sums of 2791 a converge uniformly on E Suppose that 22 a z1 is a power series with complex coef cients lfr is a positive real number and 210 loll r1 converges then Weierstrass7 observation implies that 220 a 21 converges uniformly on the closed disk consisting ofthe z E C with g 7 It follows that 220 a 21 de nes a continuous function on this closed disk lf 20 a 21 has radius of convergence R gt 0 then 20 a 21 de nes a continuous function on the open set of z E C such that lt R Math 321 Handout 33 Let V be a real vector space equipped with a norm Thus dvw H21 7 tall de nes a metric on V the metric associated to the norm Of course the real line is a one dimensional vector space and the absolute value function is a norm on R As for real numbers if 2221 uhbi are sequences of vectors in V converging to 22112 E V with respect to the metric associated to the norm then 21 Di1 converges to 21 w in V Similarly the scalar product of a convergent sequence of real numbers and a convergent sequence of vectors in V converges to the product of their limits Equivalently addition of vectors and scalar multiplication are continuous functions from V gtlt V and from R gtlt V into V lf a l is a sequence of vectors in V then the in nite series 2791 ai is said to converge in V if the sequence of partial sums 21 1 converges We say that 221 aj converges absolutely if 2791 HajH converges as a series of nonnegative real numbers If 2791 aj converges absolutely then one can check that the sequence of partial sums 2 1 ai is a Cauchy sequence as in the case of series of real numbers If V is complete with respect to the metric associated to the norm then V is said to be a Banach space By the previous remarks absolutely convergent series in a Banach space converge Conversely if every absolutely convergent series in V converges then one can show that V is complete The space CAM of bounded continuous real valued functions on a metric space M is a Banach space with respect to the supremum norm More generally if M is a metric space and V is a vector space equipped with a norm Hell then llflh Sup l x is an analogue of the supremum norm on CAM7 V for which the associated metric is the supremum metric If V is complete then CAM7 V is complete and hence a Banach space lxEM Math 321 Handout 21a Let X be a set and let A1A27 be a sequence of subsets of X The upper and lower limits of An 1 are de ned by lirnsupAn m U A n1 177 and 00 00 ligingn U H Al n1 177 By de nition lirn sup H00 An consists of the 90 E X such that 90 E An for in nitely many 71 while lirn inf H00 An consists ofthe 90 E X such that 90 E An for all but nitely many 71 which is the same as saying that 90 E An for all suf ciently large n In particular lirn inf A Q lirn sup An n oo 7Lan One might say that An 1 converges to A Q X when lirnsup H00 An lirn inf H00 An A It A Q An for each n then An il converges in this sense to A Ufa An and if An Q An for each n then A3311 converges in this sense to A 1201 An For any set E Q X let 1Ex be the indicator function of E on X which is equal to 1 when 90 E E and to 0 when 90 E XE It is easy to check that the upper and lower limits of 1A as a sequence of real numbers are the same as the indicator functions associated to the upper and lower limits of An 1 evaluated at 90 for every 90 E X respectively Math 321 Calculus Handout 1 Let M7dx7y and N7pu7v be metric spaces Suppose that E Q M and that p E M is a limit point of E Suppose also that f is a function de ned on E with values in N and that z E N We say that the limit of f as x a p with z E E is equal to 2 if for every 6 gt 0 there is a 6 gt 0 such that pf72 lt 6 lt1 for every x E E with z 31 p and d7p lt 6 Note that p may not be an element of E7 and the value of f at p is not involved in the limit even when p E E The limit is denoted gig ltz lt2 wEE when it exists It may also be denoted more simply as g fltz lt3 if E M or the choice of E is clear from the context Suppose that E M lfp is a limit point of M7 then gig fltz fp lt4 if and only if f is continuous at p If p is not a limit point of M7 then there is a 6 gt 0 such that z E M and d7p lt 6 imply that z p7 and f is automatically continuous at p

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