×

Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

or

INTRO CLASSICAL ANALYSIS

by: Evert Christiansen

20

0

3

INTRO CLASSICAL ANALYSIS MATH 615

Marketplace > Texas A&M University > Mathematics (M) > MATH 615 > INTRO CLASSICAL ANALYSIS
Evert Christiansen
Texas A&M
GPA 3.92

Natarajan Sivakumar

These notes were just uploaded, and will be ready to view shortly.

Either way, we'll remind you when they're ready :)

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

COURSE
PROF.
Natarajan Sivakumar
TYPE
Class Notes
PAGES
3
WORDS
KARMA
25 ?

Popular in Mathematics (M)

This 3 page Class Notes was uploaded by Evert Christiansen on Wednesday October 21, 2015. The Class Notes belongs to MATH 615 at Texas A&M University taught by Natarajan Sivakumar in Fall. Since its upload, it has received 20 views. For similar materials see /class/225999/math-615-texas-a-m-university in Mathematics (M) at Texas A&M University.

×

Reviews for INTRO CLASSICAL ANALYSIS

×

×

What is Karma?

You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 10/21/15
Siuakumar 1 O CO 4 0 d3 1 00 M515 080 Example Sheet 11 Suppose that f and g belong to BV1 1 Recall that HfHoo 3 sup m 3 a S 90 S b and Moo 3 SUPHWEH 3 a S 90 S b Show that fg belongs to BV1 b and Vfglmbl S HfHonglmbl HgHoonlmbl Suppose f E BV1 b Show that Vf1 b 0 if and only if f is constant throughout 11 ii Prove that the function Hme 3 WW Vfla7bl7 f E BVlaybly is a norm on BV1 b True or false lf fn is a sequence of functions such that fquot E BV1 b for every n and fn converges uniformly to f on 11 then f E BV1 b Suppose fn is a sequence of functions satisfying the following conditions a fquot E BV1 b for every n b there exists a constant A such that an1b A for every n and c fn converges pointwise to f on 11 Prove that f E BV1b and Vf1b A ii Prove that BV1b MM is a Banach space Let 1 and b be real numbers with 1 lt I Let c be a xed number in the open interval 1 1 De ne 7 0 ifa xltcg Hquot1 ifcgmgb Show that H is Riemann integrable on 11 Suppose that f is Riemann integrable on 11 and that x 2 0 for every 1 g x g I Let 12 a alt b Showthatffsz Suppose that f 11 a R satis es each of the following conditions x 2 0 for every 1 g x g 1 ii 1 is Riemann integrable on 11 and iii 1 is right continuous at every point in 1 b and left continuous at b Prove if ff 0 then x 0 for every x 6 11 Prove that the function HIH1Ow Megan is a norm on CO ii Show that C0 1 is not complete CD Suppose that a and b are real numbers with a lt I Let a g 01 lt 02 lt lt cm be xed and de ne fm1 1fmck k1n 0 otherwise 12 Show that f f O 1 ii Let T1 T2 be an enumeration of the rationals in 0119 and de ne for every n E N the function 1 1fm r1rn 0 otherwise 1W 7 b a Observe that f f 0 for every n b Find the pointwise limit of f and show that it is not Riemann integrable on 0 1 Thus the pointwise limit of a sequence of Riemann integrable functions need not be Riemann integrable 10 De ne f 01 7 R as follows ifz gcdmn1 f90 0 if z is irrational i Show that f is discontinuous at every rational number in 01 and continuous at every irrational number in 0 1 ii Show that f 0 0 De nition Suppose P is a partition of an interval 0119 The norm of P is de ned as follows maxzk 7 7671 1g k g n 11 Let f be a bounded function de ned on the compact interval 0 b b i Suppose f is Riemann integrable on 0119 and let I f Prove that for every 6 gt 0 there is a 6 66 gt 0 such that SP f 7 I lt e for everay partition P with lt 6 and any Riemann partition sum SP 1 corresponding to P ii Suppose that there exists a number I such that for every 6 gt 0 there is a 6 66 gt 0 such that SPf 7 I lt e for every partition P with lt 6 and any Riemann partition sum SP 1 corresponding to P Prove that f is Riemann integrable on 0 b and I is the Riemann integral of f 12 Suppose f is Riemann integrable on ab Prove that 1 127aquot kb7a 7 1 Jim 7 gflt T 13 Suppose that g is a nondecreasing function on 11 i Prove lf f1 and f2 are integrable with respect to 97 and f1z f2 for every a g x b7 then f1 d9 S f2 d9 ii Provealf m g g M for every a g x g b and f is integrable with respect to g on b M then mltgltbgt 7 M g fdg Mltgltbgt ego iii Prove the First Mean Value Igheorem for Riemann Stieltjes Integrals If f is continuous 12 on 11 then there is a point c in 11 such that fdg fcgb 7 9a Use ii and the Intermediate Value Theorem 14 Suppose that f E BVO7 27139 and that n is a xed positive integer Prove that S Vf 0 27139 71 02W 35 cosnm dz

×

25 Karma

×

×

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

Why people love StudySoup

Steve Martinelli UC Los Angeles

"There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

Jennifer McGill UCSF Med School

"Selling my MCAT study guides and notes has been a great source of side revenue while I'm in school. Some months I'm making over \$500! Plus, it makes me happy knowing that I'm helping future med students with their MCAT."

Jim McGreen Ohio University

"Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

Parker Thompson 500 Startups

"It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!
×

Refund Policy

STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com