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Elementary Analysis. Homework 8

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MATH5301 Elementary Analysis. Homework 8.
8.1
Show that the norms k · k1, k · kp, for p > 1 and k · k∞ are equivalent.
8.2
Let (S, k · k) and (S
0
, k · k0
) be two normed spaces. Show that the following norms on S × S
0 are equivalent.
(a) k(x, y)k1 = kxk + kyk

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MATH5301 Elementary Analysis. Homework 8.
8.1
Show that the norms k · k1, k · kp, for p > 1 and k · k∞ are equivalent.
8.2
Let (S, k · k) and (S
0
, k · k0
) be two normed spaces. Show that the following norms on S × S
0 are equivalent.
(a) k(x, y)k1 = kxk + kyk
0
(b) k(x, y)k2 =
p
kxk
2 + (kyk
0)
2
(c) k(x, y)kp = (kxk
p + (kyk
0
)
p
)
1/p
(d) k(x, y)k∞ = max{kxk, kyk
0}
8.3
Let X be a vector space and V be a normed space. The function f : X → V is called bounded if ∃M : ∀x ∈
X ⇒ kf(x)k < M. Consider the set B(X, V ) of all bounded functions from X to V .
(a) Show that B(X, V ) is a vector space.
(b) Show that the function B(X, V ) → R+:
kfk∞ := sup
x∈X
kf(x)k
defines a norm on B(X, V ) .
8.4
Let A be a dense set in a metric space (S, d), Let (V, d1) be a complete metric space and f : A → Y be an
uniformly continuous function. Show that
(a) if {xn} is a Cauchy sequence in A then {f(xn)} is a Cauchy sequence in Y .
(b) There is only one continuous function g : X → Y such that g(x) = f(x) for all x ∈ A.
8.5
Let (L, kk) be a Banach space. Let L0 be a closed subspace of L. Define the factor-space L/L0 as l1 := L/L0 =
{x + y | x ∈ L, y ∈ L0}. In other words L1 consists of all subsets of L, obtained from L0 by shifting all its
elements by some element x.
(a) Show that L1 is a vector space
(b) Define the function k · k: L1 → R+ as kxk1 = inf
x−y∈L0
kyk. Show that this function defines a norm on the
space L1.
(c) Show that L1 is a Banach space.
8.6
Let C([−1, 1]) be the space of all continuous real-valued functions f(x) with x ∈ [−1, 1]. Let kfk∞ :=
sup
x∈[−1,1]
|f(x)|. Find the distance from the point p = x
2021 to the space P2020 of all polynomials of degree
less than or equal to 2020.