QUIZ I
MTL 411 FUNCTIONAL ANALYSIS
The i-th question carries i marks, where i ∈ {1, 2, 3, 4, 5}.
Maximum marks that can be earned is 10.
1. Let X and Y be normed linear spaces and A : X → Y be a linear map. If A is a bounded
map, then prove that (Axn )n∈N is a Cauchy sequence in Y , whenever (xn )n∈N is a Cauchy
sequence in X.
2. Let f be a linear functional on a normed linear space X and let x0 ∈
/ N (f ), the null space of
f . Show that any x in X has a unique representation as
x = y + αx0
with α ∈ K and y ∈ N (f ).
3. Prove or disprove the following statement:
Any two normed linear spaces having same finite dimension are linearly homeomorphic.
4. Let X = C[0, 1] be endowed with the sup-norm. Define T : X → X by
∫ t
x 7→ T x(= y), where y(t) = x(u) du.
0
(i) Is T a bounded linear operator ?
(ii) Describe Rg(T ), the range space of T .
(iii) Is T −1 : Rg(T ) → X a well defined bounded linear operator ?
5. Prove that the converse of the assertion in question 1 is also true. That is, a linear map
A : X → Y is bounded if and only if A maps Cauchy sequences in X into Cauchy sequences
in Y .
1
MTL 411 FUNCTIONAL ANALYSIS
The i-th question carries i marks, where i ∈ {1, 2, 3, 4, 5}.
Maximum marks that can be earned is 10.
1. Let X and Y be normed linear spaces and A : X → Y be a linear map. If A is a bounded
map, then prove that (Axn )n∈N is a Cauchy sequence in Y , whenever (xn )n∈N is a Cauchy
sequence in X.
2. Let f be a linear functional on a normed linear space X and let x0 ∈
/ N (f ), the null space of
f . Show that any x in X has a unique representation as
x = y + αx0
with α ∈ K and y ∈ N (f ).
3. Prove or disprove the following statement:
Any two normed linear spaces having same finite dimension are linearly homeomorphic.
4. Let X = C[0, 1] be endowed with the sup-norm. Define T : X → X by
∫ t
x 7→ T x(= y), where y(t) = x(u) du.
0
(i) Is T a bounded linear operator ?
(ii) Describe Rg(T ), the range space of T .
(iii) Is T −1 : Rg(T ) → X a well defined bounded linear operator ?
5. Prove that the converse of the assertion in question 1 is also true. That is, a linear map
A : X → Y is bounded if and only if A maps Cauchy sequences in X into Cauchy sequences
in Y .
1
