Unformatted text preview: [0 CI . Assume that. the family {fn}°° of realvalued functions on [0, 1] is equicontii‘niwis a: Graduate Proﬁciency Examination
Analysis Fall 1993 Instructions: Do all problems. Each problem is Worth 10 points. . Given a Cl function F : R" —+ R” satisfying; lF(1I=) E3 IIH‘HQa 2‘ E 1R", prove that there is an e > 0 such that the equation F (T) = m + a has a solution 3:
whenever the vector a satisfies ”a” < 6. 00
. If an 2 0 and 2 an < oo prove that there exists a sequence 1),, such that lim 1),, = 71:] n—~oo M)
+00 and Z a,,bn converges.
n=1 ..1 ”=1
. . t b . a ,
pomtvmse bounded. Also assume [a f,,(.1r)d.r —> 0 as n ——> (x; for every U S a <: I) f: l.
Prove that f" —~> 0 uniformly. k . Let P17; denote the set of realvalued polynomials which involve no odd powers of the variable, i.e., the coefficient of each odd power term is zero. Prove that PE is dense in C ([0, 1]) with the sup norm. For which closed intervals other than [0,1] can the same
be proved? . For which non—decreasing functions 5 on [0.1] does the RieinannStieltjes integral 1 . , .
f0 ,Bdﬂ exxst? Prove your assertion. . If f is continuous and lim f(s) 2: a, prove that —1— t f—(sﬂ (1.3 —+ a as t —i 9c. 3—00 logt 1 ...
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 Spring '11
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