8/18/2019 Math 104 UC Berkeley Spring 2016 HW 4
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HOMEWORK 13 (DUE APRIL 28TH)
1) Ross 26.8
2) Find an explicit formula for
∞
n=0 n3xn for |x| < 1.
3) Ross 13.1
4) Ross 13.10
5) We define a map f : (S, d) → (S , d) between 2 metric spaces to be continuous at x ∈ S if for every sequence {xn} ⊂ S that converges to x, the sequence {f (xn)} ⊂S is convergent to f (x).
a) Given a metric space (S,d) and a fixed point x0 ∈ S, define f : (S, d) →
(R, dstd) to bef (x) := d(x, x0).
Show that d is a continuous map.b) Given a metric space (S,d) and a fixed bounded subset A ⊂ S , define f :
(S,d) → (R, dstd)f (x) := inf y∈Ad(x, y).
Show that f is continuous.
6) Let C 0([a, b]) denote the space of continuous functions f : [a, b] → R. Define
d(f, g) = sup[a,b]|f − g|.
a) Show that d is a metric in C 0([a, b]).b) Is this metric space complete, i.e. any Cauchy sequence in C 0([a, b]) converges
to an element in C 0([a, b]).?c) Define F : C 0([a, b]) → R to be
F (f ) =
ba
f.
Show that F is a continuous map.
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