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.
