April 2008 MATH 321 Name Page 2 of 13 pages
Marks
[9] 1. Define
(a) uniform convergence of a sequence of functions
(b) an algebra of functions that vanishes nowhere
(c) an atlas and a maximal atlas
Continued on page 3
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Atlas - Mathematics - Exam, Exams of Mathematics
This is the Exam of Mathematics which includes Continuous, Number, Every, Domain, Continuous Functions, Laplace Transforms, Constant, Simplify, Evaluate, Value of the Constant etc. Key important points are: Atlas, Uniform Convergence, Sequence of Functions, Algebra of Functions, Vanishes, Maximal Atlas, Example, Riemann Integrable, Function, Converge
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2012/2013
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Marks
[9] 1. Define (a) uniform convergence of a sequence of functions (b) an algebra of functions that vanishes nowhere (c) an atlas and a maximal atlas
[16] 2. Give an example of each of the following, together with a brief explanation of your example. If an example does not exist, explain why not. (a) a function f : [0, 1] → IR which is Riemann integrable on [0, 1] but for which the function F : [0, 1] → IR defined by F (x) = ∫^0 x f (t) dt is not Riemann integrable on [0, 1] (b) a sequence of functions that converges to zero pointwise on [0, 1] and uniformly on [ε, 1 −ε] for every ε > 0, but does not converge uniformly on [0, 1] (c) a Fourier series (^) n=∑∞−∞ cneinx^ that does not converge in the mean (d) two charts for (− 1 , 1) (with the usual metric) that are not compatible
[15] 3. Let α, f, g : [a, b] → IR with α an increasing function. (a) Prove that ∫¯ (^) b a^ (f^ +^ g)^ dα^ ≤^
∫¯ (^) b a^ f dα^ +^
∫¯ (^) b a^ g dα (b) Either prove that ∫¯ (^) b a^ (f^ +^ g)^ dα^ =^
∫¯ (^) b a^ f dα^ +^
∫¯ (^) b a^ g dα or provide a counterexample.
[15] 5. Let {fn^ } n∈IN be a uniformly convergent sequence of continuous real–valued functions defined on a metric space M and let g be a continuous function on IR. Define, for each n ∈ IN, hn(x) = g(fn(x)). (a) Let M = [0, 1]. Prove that the sequence {hn^ } n∈IN converges uniformly on [0, 1]. (b) Let M = IR. Either prove that the sequence {hn^ } n∈IN converges uniformly on IR or provide a counterexample.
[15] 7. Let α > 0. A function f : IR → IR is said to be H¨older continuous of exponent α if the quantity
‖f ‖α = sup x 6 =y^ |f^ ( |xx)−−yf|^ α(y)| is finite. Let {fn^ } n∈IN be a sequence of H¨older continuous real valued functions on IR that obey supx∈IR |fn(x)| ≤ 1 and ‖fn‖α ≤ 1 for all n ∈ IN. Prove that there is a continuous function f : IR → IR and a subsequence of {fn^ } n∈IN that converges pointwise to f and that furthermore converges uniformly to f on [−M, M ] for every M > 0.
Be sure that this examination has 13 pages including this cover
The University of British Columbia Sessional Examinations - April 2008 Mathematics 321 Real Variables II
Closed book examination Time: 2 12 hours
Name Signature Student Number Instructor’s Name Section Number
Special Instructions:
No calculators, notes, or other aids are allowed.
Rules Governing Formal Examinations
- Each candidate must be prepared to produce, upon request, a UBCcard for iden- tification.
- Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.
- No candidate shall be permitted to enter the examination room after the expiration of one half hour from the scheduled starting time, or to leave during the first half hour of the examination.
- Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action: (a) having at the place of writing any books, papers or memoranda, calculators, computers, sound or image players/recorders/transmitters (including telephones), or other memory aid devices, other than those authorized by the examiners; (b) speaking or communicating with other candidates; and (c) purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness shall not be received.
- Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any examination material from the examination room without permission of the invigilator.
- Candidates must follow any additional examination rules or directions communi- cated by the instructor or invigilator.
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