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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