{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# Analysis1993Aug - [0 CI Assume that the family{fn}°° of...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: [0 CI . Assume that. the family {fn}°° of real-valued 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; l|F(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 real-valued 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 Rieinann-Stieltjes 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 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online