Functional Analysis

ASSIGNMENT SHEET I

MTL 411 FUNCTIONAL ANALYSIS



1. Let X be a normed linear space and X0 be a subspace of X. Show that for every x ∈ X,
u ∈ X0 and α ∈ K,
dist(αx, X0 ) = |α| dist(x, X0 ), dist(x + u, X0 ) = dist(x, X0 ).

2. Let X be a normed linear space and X0 be a subspace of X. Then show that
p(x) = dist(x, X0 ) := inf{∥x − u∥ : u ∈ X0 }
defines a seminorm on X. Is it a norm ? - Why ?

3. For x : [0, 1] → K, let ν(x) = |x(0)| + |x( 12 )| + |x(1)|. Check whether ν is a norm on the space
(a) C[0, 1], (b) P2 [0, 1].

4. Let C0 (R) be the set of all continuous functions x : R → K such that |x(t)| → 0 as |t| → ∞.
Prove that C0 (R) is a closed subspace of B(R), the space of all bounded K-valued functions
defined on R, endowed with the supnorm. Is C0 (R) a Banach space ?

5. Let X be a normed linear space with norm ∥.∥. Define
d(x, y) = min{1, ∥x − y∥} ∀ x, y ∈ X.
Prove that d is a metric on X which is not induced by any norm on X.

6. Prove that c00 ⊆ l1 ⊆ lp ⊆ lr ⊆ c0 ⊆ c ⊆ l∞ for every p, r ∈ (1, ∞) with p < r. Also prove
that the inclusions are strict.

7. Let F(N) denote the linear space of all K-valued functions defined on N with respect to the
∑∞ 1
standard operations. Prove∑that for 0 < p < 1, ∥x∥p := [ i=1 |x(i)|p ] p does not define a
∞
norm on X := {x ∈ F(N) : i=1 |x(i)|p < ∞}.

8. For x ∈ C 1 [0, 1], the space of all continuously differentiable functions, define
∥x∥∗ = |x(0)| + ∥x′ ∥∞ .
Show that
(i) x 7→ ∥x∥∗ is a norm on C 1 [0, 1],
(ii) C 1 [0, 1] endowed with ∥.∥∗ is a Banach space.
Is ∥.∥∗ equivalent to the norm ∥.∥∞ on C 1 [0, 1]?- Why?

9. Let C k [a, b], k ≥ 1, denote the space of all k-times continuously differentiable functions defined
on the interval [a, b]. Show that C k [a, b] is an incomplete normed linear space under the
supnorm. Further show that C k [a, b] is a Banach space under the norm

∑
k
∥x∥ = ∥x(m) ∥∞ , where ∥x(m) ∥∞ = max |x(m) (t)|, x(0) = x.
t∈[a,b]
m=0

10. Show that equivalent norms on a linear space X induce the same topology for X.
∑n
11. For 1 ≤ p < ∞ and x ∈ lp , define xn = j=1 x(j)ej , n ∈ N. Show that the sequence (xn )
converges to x with respect to the norm ∥.∥p . Is {e1 , e2 , . . . } a basis for l1 ? -Why ?

12. For p ∈ [1, ∞), show that the norm ∥.∥p on C[a, b] is not equivalent to ∥.∥∞ .
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