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Unformatted text preview: £02: M421 HW 5 a»; Due Friday NOV. 9 From Wade Section Page Number Problems
11.1 329330 4, 5 11.2 337—339 3, 4, 5, 6, 7 Nonbook Exercises
1) For which oz > 0 is the function $2340
f($,y)={0;21l_1yL3 ($,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], foo g Mlf1,1.
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
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 Fall '07
 PROMISLOW
 Topology, $1, #, Cauchy sequence, Section Page Number

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