Department of Mathematics
Indian Institute of Technology Delhi
MTL 411 Functional Analysis: Minor Test I
Maximum Marks: 25 Time: 1 hour
1. Write your answers coherently, stating clearly the relevant results.
2. Each question carries 5 marks. Maximum that can be earned is 25.
_________________________________________________________________________________________
1. Let (X, ∥.∥) be a normed linear space and consider the open unit ball U (0, 1) := {x ∈ X : ∥x∥ < 1}.
Prove that the map f : X → R defined by
( ) { }
f (x) = dist x, U (0, 1) := inf ∥x − u∥ : u ∈ U (0, 1)
is uniformly continuous. If X = R2 is endowed with the l1 -norm defined by
( )
∥x∥ = |x(1)| + |x(2)|; x = x(1), x(2) ∈ R2 ,
describe the kernel of the map f , namely, {x ∈ R2 : f (x) = 0}. [3 + 2 = 5]
2. Consider C[0, 1], the space of all continuous real valued functions, endowed with the supnorm
{ }
∥x∥∞ := sup |x(t)| : t ∈ [0, 1] .
Prove or disprove the following assertions:
{ }
(a) A := x ∈ C[0, 1] : x is injective is closed.
{ }
(b) B := x ∈ C[0, 1] : x( 12 ) = 0 endowed with the supnorm is a Banach space.
[2 + 3 = 5]
( )
3. Let X = C[0, 1], ∥.∥∞ and Y = P2016 [0, 1], the set of all polynomials of degree not exceeding 2016
with real coefficients defined on [0, 1]. Does there exist:
∑2016
(a) a polynomial y ∗ (t) = k=0 a∗k tk satisfying
∑
2016 ∑
2016
sup et − a∗k tk ≤ sup et − ak tk ∀ ai ∈ R.
t∈[0.1] k=0 t∈[0.1] k=0
(b) a continuous function x such that
sup |x(t)| = 1 and inf sup |x(t) − y(t)| = 1.
t∈[0,1] y∈P2016 [0,1] t∈[0,1]
Justify. [3 + 2 = 5]
Indian Institute of Technology Delhi
MTL 411 Functional Analysis: Minor Test I
Maximum Marks: 25 Time: 1 hour
1. Write your answers coherently, stating clearly the relevant results.
2. Each question carries 5 marks. Maximum that can be earned is 25.
_________________________________________________________________________________________
1. Let (X, ∥.∥) be a normed linear space and consider the open unit ball U (0, 1) := {x ∈ X : ∥x∥ < 1}.
Prove that the map f : X → R defined by
( ) { }
f (x) = dist x, U (0, 1) := inf ∥x − u∥ : u ∈ U (0, 1)
is uniformly continuous. If X = R2 is endowed with the l1 -norm defined by
( )
∥x∥ = |x(1)| + |x(2)|; x = x(1), x(2) ∈ R2 ,
describe the kernel of the map f , namely, {x ∈ R2 : f (x) = 0}. [3 + 2 = 5]
2. Consider C[0, 1], the space of all continuous real valued functions, endowed with the supnorm
{ }
∥x∥∞ := sup |x(t)| : t ∈ [0, 1] .
Prove or disprove the following assertions:
{ }
(a) A := x ∈ C[0, 1] : x is injective is closed.
{ }
(b) B := x ∈ C[0, 1] : x( 12 ) = 0 endowed with the supnorm is a Banach space.
[2 + 3 = 5]
( )
3. Let X = C[0, 1], ∥.∥∞ and Y = P2016 [0, 1], the set of all polynomials of degree not exceeding 2016
with real coefficients defined on [0, 1]. Does there exist:
∑2016
(a) a polynomial y ∗ (t) = k=0 a∗k tk satisfying
∑
2016 ∑
2016
sup et − a∗k tk ≤ sup et − ak tk ∀ ai ∈ R.
t∈[0.1] k=0 t∈[0.1] k=0
(b) a continuous function x such that
sup |x(t)| = 1 and inf sup |x(t) − y(t)| = 1.
t∈[0,1] y∈P2016 [0,1] t∈[0,1]
Justify. [3 + 2 = 5]
