HW5-PA - £02: M421 HW 5 a»; Due Friday NOV. 9 From Wade...

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Unformatted text preview: £02: M421 HW 5 a»; Due Friday NOV. 9 From Wade Section Page Number Problems 11.1 329-330 4, 5 11.2 337—339 3, 4, 5, 6, 7 Non-book Exercises 1) For which oz > 0 is the function $2340 f($,y)={0;21l_1yL|3 ($,y)aé0 (at, y) = 0, differentiable at zero? 2) Consider R°° = {f2 ($1,$2,$3, . . x,- E R,i=1,2,3,...}. The l2 norm on R°° is co % “5H2 = (Elm?) . i=1 The space I2 = {50' E R°°| < 00}, is infinite dimensional. Show that the l2 unit sphere, S = {50' 6 l2! = 1}, is closed, bounded, and not sequentially compact. That is, find a sequence from S which has no convergent subsequence. Honor’s Problems 3) Define the space C1[a,b] = {f : [a, b] r—> R‘ fandf’ E C[a,b]}, and the norm b llfllm =/(|f(x)|+|f’(m)|)dm. a (a) Show there exists M > 0 such that for all f E Cl[a,b], ||f||oo g Ml|f||1,1. Hint: Use the Fundamental Theorem 0f Calculus. (b) Show that if {fn}g°=1 is a sequence from C1 and fn —> g in 0 “1,1 then fn —> 9 point wise. (0) Define Wl’1 to be the set of all sequences from C1 which are cauchy in the norm 0 “1,1. Show that for any sequence {fn} from C'1 which is cauchy in H 0 “1,1 there is a g E C[a, b] such that _.9H1 —’ 0- For this reason we say that W” c C[a, b]. 4)(a) Let f E L1[a, b] and g E W1’1[a,b]. Show that the product fg E L1[a, b]. That is, if f is represented by the H 0 “1 cauchy sequence {fn} C C [(1, b] and g by the 0 “1,1 cauchy sequence {gn} C Cl[a, b], then the sequence {hn} where hn = fngn is contained in C[a, b] and is cauchy in ll 0 Hi- (b) In part (a), show that if g is merely in L1[a, b], then the product f 9 may not be in L1[a, b]. That is find two sequences {fn} and {gn}, both from C[a, b] and both cauchy in H 0 “1 such that the “product” {fngn} is not cauchy in o For who °<>0 r5 ‘ XLM'" pm : 3% XQHXI” O ASA” Wha‘ah a} 1 °( 00 cm ' 4M & koc-\ Vlkl ivy) : Mum M V\ *Hz‘ 0 90‘ Q m w} 0W5?» Q4 em‘aaW # {ad is Vsj‘ Jiufia ‘75) v?» is m4 Wrib 2:5. 099‘ S S MI )N‘HAJ " gws NB“? ’ A X W Cws'aém “loo: 91 CM.” A4,”) \ mm CALL"), g Lu ‘3 ‘00 "L “Hz :(21 “(3‘) "N Spud? 9}; 3mm“! imam”) r3 minds JamsmO 3 \\“wi\\~\\?<\l\—so .33 Ma fing- go (ifs (Abs/QC» CW 3:55:14, but 1. “‘6er Ln} 5“: ¢% (0... ol‘lol lam] \ \ a 9 q .5 OKCOV\\Y6910\°M 04) a“ an; :3 Yul\awi‘-Q“Qzu ’; ligand“ ...
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This note was uploaded on 07/25/2008 for the course MATH 421 taught by Professor Promislow during the Fall '07 term at Michigan State University.

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HW5-PA - £02: M421 HW 5 a»; Due Friday NOV. 9 From Wade...

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