IGNOU MMT 6 SOLVED ASSIGNMENT

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MMT 6 2025 - English

Course Code: MMT-006

Assignment Code: MMT-006/TMA/2025

Maximum Marks: 100

1. State whether the following statements True or False? Justify your answers:

a) The function  defined on  as:

b) C_o is a Banach space.

c) If A is the right shift operator on l², then the eigen spectrum is non-empty.

d) If a normed linear space is reflexive, then so is its dual space.

e) If a normed linear space X is finite dimensional, then so is X'.

2. a) Consider the space , define . Show that f is a linear functional which is not continuous w.r.t the norm

b) Consider the space C¹[0,1] of all C¹ functions on [0,1] endowed with the uniform norm induced from the space C[0,1], and consider the differential operator defined by Df = f'. Prove that D is linear, with closed graph, but not continuous. Can we conclude from here that C¹[0, 1] is not a Banach space? Justify your answer.

3. a) When is a normed linear space called separable? Show that a normed linear space is separable if its dual is separable [You should state all the proposition or theorems or corollaries used for proving the theorem]. Is the converse true? Give justification for your answer. [Whenever an example is given, you should justify that the example satisfies the requirements.]

b) Let X be a Banach space, Y be a normed linear space and be a subset of B(X, Y). If is not uniformly bounded, then there exists a dense subset D of X such that for every is not bounded in Y.

4. a) Read the proof of the closed graph theorem carefully and explain where and how we have used the following facts in the proof.

i) X is a Banach space.

ii) Y is a Banach space.

iii) F is a closed map.

iv) Which property of continuity is being established to conclude that F is continuous.

b) Which of the following maps are open? Give reasons for your answer.

i)

ii)

5. a) Let f: C[0,1]→ be given by f (x) = x(1)∀x ∈ C[0,1]. Show that f is continuous w.r.t the supnorm and f is not continuous w.r.t the p-norm.

b) Let X be an inner product space and x, y ∈ X. Prove that x | y if and only if

6. a) Let H=R³ and F be the set of all x = (x₁, x₂, x₃) in H such that x₁ = 0. Find F¹. Verify that every x ∈ H can be expressed as x = y + z where y ∈ Fand z ∈ F¹.

b) Given an example of an Hilbert space H and an operator A on Η such that σ_e(A)is empty. Justify your choice of example.

c) Let A be a normal operator on a Hilbert space X. Show that σ(A) ⊂ σ_a(A) where σ_a (A) denotes the approximate eigen spectrum of A and σ(A) denotes the spectrum of A.

7. a) Let X = C₀₀ with  Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example.

b) Give one example of each of the following. Also justify your choice of example.

i) A self-adjoint operator on .

ii) A normal operator on a Hilbert space which is not unitary.

c) Let X be a normed space and Y be proper subspace of X. Show that the interior Yº of Y is empty.

8. a) Let X,Y be normed spaces and suppose BL(X,Y) and CL(X,Y) denote, respectively, the space of bounded linear operators from X to Y and the space of compact linear operators from X to Y. Show that CL(X,Y) is linear subspace of BL(X,Y). Also, Show that if Y is a Banach space, then CL(X,Y) is a closed subspace of BL(X,Y).

b) Define a Hilbert-Schmidt operator on a Hilbert space H and give an example. Is every Hilbert-sehmidt operator a compact operator? Justify your answer.

9. a) Let {A_n} be a sequence of unitary operators in BL(H). Prove that if , then A is unitary.

b) Define the spectral radius of a bounded linear operator A ∈ BL(X). Find the spectral radius of A in BL, where A is given by the matrix

with respect to the standard basis of .

c) Let X be a Banach space and Y be a closed subspace of X. Let π: X → X/Y be canonical quotient map. Show that is open.

10. a) Give an example of a compact linear map on l².

b) Give an example of a positive operator on .

c) Prove the following result:

Suppose A is a non-zero compact self-adjoint operator on a Hilbert space H over K. Prove that there exists a finite set {r₁,r₂,..., r_n} of a non-zero real numbers with  and an orthonormal set {w₁, w₂,..., w_n} in H such that

Further, mention in which step of the proof it is used that A is a compact self-adjoint operator. Explain why?

MMT 6 2026 - English

Assignment

Course Code: MMT-006

Assignment Code: MMT-006/TMA/2026

Maximum Marks: 100

1. State whether the following statements True or False? Justify your answers:

a) The function defined on as:
for
is a norm.

b) C₀ is a Banach space.

c) If A is the right shift operator on l², then the eigen spectrum is non-empty.

d) If a normed linear space is reflexive, then so is its dual space.

e) If a normed linear space X is finite dimensional, then so is X'.

2. a) Consider the space c₀₀. For , define . Show that f is a linear functional which is not continuous w.r.t the norm .

b) Consider the space C¹[0,1] of all C¹ functions on [0,1] endowed with the uniform norm induced from the space C[0,1], and consider the differential operator defined by . Prove that D is linear, with closed graph, but not continuous. Can we conclude from here that C¹[0,1] is not a Banach space? Justify your answer.

3. a) When is a normed linear space called separable? Show that a normed linear space is separable if its dual is separable [You should state all the proposition or theorems or corollaries used for proving the theorem]. Is the converse true? Give justification for your answer. [Whenever an example is given, you should justify that the example satisfies the requirements.]

b) Let X be a Banach space, Y be a normed linear space and be a subset of B(X, Y). If is not uniformly bounded, then there exists a dense subset D of X such that for every is not bounded in Y.

4. a) Read the proof of the closed graph theorem carefully and explain where and how we have used the following facts in the proof.

i) X is a Banach space.

ii) Y is a Banach space.

iii) F is a closed map.

iv) Which property of continuity is being established to conclude that F is continuous.

b) Which of the following maps are open? Give reasons for your answer.

i) given by .

ii) given by .

5. a) Let be given by . Show that f is continuous w.r.t the supnorm and f is not continuous w.r.t the p-norm.

b) Let X be an inner product space and . Prove that if and only if .

6. a) Let and F be the set of all in H such that . Find . Verify that every can be expressed as where and . 

b) Given an example of an Hilbert space H and an operator A on H such that is empty. Justify your choice of example. 

c) Let A be a normal operator on a Hilbert space X. Show that where denotes the approximate eigen spectrum of A and denotes the spectrum of A. 

7. a) Let with . Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example. 

b) Give one example of each of the following. Also justify your choice of example. 
i) A self-adjoint operator on .
ii) A normal operator on a Hilbert space which is not unitary.

c) Let X be a normed space and Y be proper subspace of X. Show that the interior Y⁰ of Y is empty. 

8. a) Let X, Y be normed spaces and suppose BL(X, Y) and CL(X, Y) denote, respectively, the space of bounded linear operators from X to Y and the space of compact linear operators from X to Y. Show that CL(X, Y) is linear subspace of BL(X, Y). Also, Show that if Y is a Banach space, then CL(X, Y) is a closed subspace of BL(X, Y). 

b) Define a Hilbert-Schmidt operator on a Hilbert space H and give an example. Is every Hilbert-schmidt operator a compact operator? Justify your answer. 

9. a) Let be a sequence of unitary operators in BL(H). Prove that if , then A is unitary.

b) Define the spectral radius of a bounded linear operator . Find the spectral radius of A in , where A is given by the matrix

with respect to the standard basis of .

c) Let X be a Banach space and Y be a closed subspace of X. Let be canonical quotient map. Show that is open.

10. a) Give an example of a compact linear map on l².

b) Give an example of a positive operator on .

c) Prove the following result:

Suppose A is a non-zero compact self-adjoint operator on a Hilbert space H over K. Prove that there exists a finite set of a non-zero real numbers with and an orthonormal set in H such that

Further, mention in which step of the proof it is used that A is a compact self-adjoint operator. Explain why?

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