## Description

MATH5301 Elementary Analysis. Homework 9

9.1

Let k · ka and k · kb be two equivalent norms on R

n.

(a) Prove that if the set A is closed in the a-norm, then it is closed in b-norm.

(b) Prove that if the set A is compact in the a-norm, then it is compact in b-norm.

9.2

Consider the set `∞ of all real-valued sequences, endowed with the sup-norm: klk∞ = sup

n∈N

|ln|.

(a) Prove that `∞ is complete.

(b) Prove that `∞ is not compact.

9.3

Consider the set B([0, 1], R) of all bounded real-valued functions on the unit interval, endowed with the sup-norm:

kfk∞ = sup

x∈[0,1]

|f(x)|. Denote by B1 := {f ∈ B : kfk∞ 6 1} be close unit ball.

(a) Prove B1 is closed.

(b) Prove that B1 is bounded.

(c) Prove that B1 is not compact.

9.4

Let {V, k · k} be a normed space. Show that the function f(x) = kxk : V → R is continuous on V .

9.5

Let (X, d1) and (Y, d2) are two metric spaces. Assume also that Y is a vector space. Construct an example of

two continuous functions f, g : X → Y such that f + g is discontinuous.

9.6

Construct an example of a sequence {fn} of nowhere continuous functions [0, 1] → R such that fn converge in

sup-norm to continuous function.