IGNOU MSCMACS MMT 6 Functional Analysis Solved Assignment English

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MSCMACS 2026 - English

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MMT 6: Functional Analysis

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Title Name	IGNOU MSCMACS MMT 6 Functional Analysis Solved Assignment English
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MSCMACS
Course Name	M.Sc. Mathematics with Applications in Computer Science
Subject Code	MMT 6
Subject Name	Functional Analysis
Year	2026
Session	-
Language	English
Assignment Code	MMT 6/Assignment-1/2026
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2026. Latest MMT 06 2026 Solved Assignment Solutions
Format	Ready-to-Print PDF (.soft copy)

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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: $5 \times 2 = 10$

a) The function $\|\cdot\|$ defined on $\mathbb{R}^n$ as:
$\|x\| = \sum_{j=1}^{n} |a_j|$ for $x = (a, a_2, \dots, a_n) \in \mathbb{R}^n$
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 $x = (x_1, x_2, \dots, x_n, \dots) \in c_{00}$ , define $f(x) = \sum_{n=1}^{\infty} x_n$ . Show that f is a linear functional which is not continuous w.r.t the norm $\|x\| = \sup_n |x_n|$ .

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 $D : (C^1[0,1], \|\cdot\|_\infty) \rightarrow (C[0,1], \|\cdot\|_\infty)$ 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 $\mathcal{F}$ be a subset of B(X, Y). If $\mathcal{F}$ is not uniformly bounded, then there exists a dense subset D of X such that for every $x \in D, \{F(x) : F \in \mathcal{F}\}$ 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) $T : \mathbb{R}^3 \to \mathbb{R}^2$ given by $T(x,y,z) = (x,z)$ .

ii) $T : \mathbb{R}^3 \to \mathbb{R}^3$ given by $T(x,y,z) = (x,y,0)$ .

5. a) Let $f : C[0,1] \to \mathbb{R}$ be given by $f(x) = x(1) \forall x \in C[0,1]$ . 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 $x,y \in X$ . Prove that $x \perp y$ if and only if $\|kx+y\|^2 = \|kx\|^2 + \|y\|^2, k \in K$ .

6. a) Let $H = \mathbb{R}^3$ and F be the set of all $\mathbf{x} = (x_1, x_2, x_3)$ in H such that $x_1 = 0$ . Find $F^\perp$ . Verify that every $\mathbf{x} \in H$ can be expressed as $\mathbf{x} = \mathbf{y} + \mathbf{z}$ where $\mathbf{y} \in F$ and $\mathbf{z} \in F^\perp$ .

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

c) Let A be a normal operator on a Hilbert space X. Show that $\sigma(A) \subseteq \sigma_a(A)$ where $\sigma_a(A)$ denotes the approximate eigen spectrum of A and $\sigma(A)$ denotes the spectrum of A.

7. a) Let $X = c_{00}$ with $\|\cdot\|_p$ . 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 $\ell^2$ .
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 $\{A_n\}$ be a sequence of unitary operators in BL(H). Prove that if $\|A_n - A\| \to 0, A \in BL(H)$ , then A is unitary.

b) Define the spectral radius of a bounded linear operator $A \in BL(X)$ . Find the spectral radius of A in $BL(\mathbb{R}^3)$ , where A is given by the matrix

$$ \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix}$

with respect to the standard basis of $\mathbb{R}^3$ .

c) Let X be a Banach space and Y be a closed subspace of X. Let $\pi : X \to X/Y$ be canonical quotient map. Show that $\pi$ is open.

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

b) Give an example of a positive operator on $(\mathbb{C}^n, \|\cdot\|_2)$ .

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 $\{r_1, r_2, \dots, r_n\}$ of a non-zero real numbers with $|r_1| \ge |r_2| \ge |r_3| \ge \dots \ge |r_n|$ and an orthonormal set $\{w_1, w_2, \dots, w_n\}$ in H such that

$$ A(x) = \sum_{i=1}^{n} r_i \langle x, w_i \rangle w_i, \quad x \in H.$

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

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 $mathbb{R}_{n}$ as:

$|x|=sum_{j=1}^{n}|a_j|quad ext{for}x=(a_1,a_2,dots,a_n)inmathbb{R}^n$

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 $c_{00}. ext{For}x=(x_1,x_2,dots,x_n,dots)in c_{00}$ , define $f(x)=sum_{n=1}^{infty}x_n$ . Show that f is a linear functional which is not continuous w.r.t the norm $|x|=sup_n|x_n|$

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 $D:(C^1[0,1],|cdot|_infty) o(C[0,1],|cdot|_infty)$ 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.

b) Let X be a Banach space, Y be a normed linear space and $mathcal{F}$ be a subset of B(X, Y). If $mathcal{F}$ is not uniformly bounded, then there exists a dense subset D of X such that for every $xin D,{F(x):Finmathcal{F}}$ 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) $T:mathbb{R}^3 omathbb{R}^2 ext{given by}T(x,y,z)=(x,z).$

ii) $T:mathbb{R}^3 omathbb{R}^3 ext{given by}T(x,y,z)=(x,y,0).$

5. a) Let f: C[0,1]→ $mathbb{R}$ 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

$|kx+y|^2=|kx|^2+|y|^2,quad kin K.$

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 $|-|_p$ 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 $ell^2$ .

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 $|A_n-A| o 0,quad Ain BL(H)$ , 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 $(mathbb{R}^3)$ , where A is given by the matrix

$egin{bmatrix}0&1&0\-1&0&0\0&0&-1end{bmatrix}$

with respect to the standard basis of $mathbb{R}^{3}$ .

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 $(C^n,|cdot|_2)$ .

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 $|r_1|ge|r_2|ge|r_3|gedotsge|r_n|$ and an orthonormal set {w₁, w₂,..., w_n} in H such that

$A(x)=sum_{i=1}^{n}r_ilangle x,w_i angle w_i,quad xin H.$

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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Course: M.Sc. Mathematics with Applications in Computer Science (MSCMACS)
Subject: MMT 6 - Functional Analysis
Session: January 2026 - December 2026
Language: English

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Product Information

Course Name	M.Sc. Mathematics with Applications in Computer Science
Course Code	MSCMACS
Subject Code	MMT 6
Subject	Functional Analysis
Programme	MASTER DEGREE PROGRAMMES Courses
Language	English
Material Type	Solved Assignment – Soft Copy / PDF

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