IGNOU MMT-006 Solved Assignment 2024 | M.Sc. MACS

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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IGNOU MMT-006 Assignment Question Paper 2024

mmt-006-assignment-question-paper-aab71826-936e-4140-892f-f2b7ca3c6de5

a) Let $X = {f \in C [0, 1] : f (0) = 0}$

Y = {g \in x : \int_{0}^{1} g (t) d t = 0}

Prove that

Y

is a proper subspace of

X

. Is

Y

a closed subspace of

X

? Justify your answer.
b) Let

X = L^{p} [0, 1]

and

x = x (t) = t^{2}

. Find

∥ x ∥_{p}

for

p = 4

and

\infty

.
c) Let E be a subset of a normed space

X, Y = span E

and

a \in X

. Show that

a \in \bar{Y}

if and only if

f (a) = 0

whenever

f \in X^{'}

and

f = 0

everywhere on

E

.
2. a) Consider the space

c_{00}

. 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

C^{1} [0, 1]

of all

C^{1}

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}) \to (C [0, 1], ∥ \cdot ∥_{\infty})

defined by

D f = f^{'}

. Prove that

D

is linear, with closed graph, but not continuous. Can we conclude from here that

C^{1} [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

F

be a subset of

B (X, Y)

. If

F

is not uniformly bounded, then there exists a dense subset

D

X

such that for every

x \in D, {F (x) : F \in 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 : R^{3} \to R^{2}

given by

T (x, y, z) = (x, z)

.
ii)

T : R^{3} \to R^{3}

given by

T (x, y, z) = (x, y, 0)

.
5. a) Let

f : C [0, 1] \to 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 ⊥ y

if and only if

∥ k x + y ∥^{2} = ∥ k x ∥^{2} + ∥ y^{2} ∥, k \in K

.
6. a) Let

H = R^{3}

and

F

be the set of all

x = (x_{1}, x_{2}, x_{3})

H

such that

x_{1} = 0

. Find

F^{⊥}

. Verify that every

x \in H

can be expressed as

x = y + z

where

y \in F

and

z \in F^{⊥}

.
b) Given an example of an Hilbert space

H

and an operator

A

H

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) \subset 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 = 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

ℓ^{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

Y^{0}

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

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

C L (X, Y)

is a closed subspace of

B L (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

B L (H)

. Prove that if

∥ A_{n} - A ∥ \to 0, A \in B L (H)

, then

A

is unitary.
b) Define the spectral radius of a bounded linear operator

A \in B L (X)

. Find the spectral radius of

A

B L (R^{3})

, where

A

is given by the matrix

[\begin{array}{rrr} 0 & 1 & 0 \\ - 1 & 0 & 0 \\ 0 & 0 & - 1 \end{array}]

with respect to the standard basis of

R^{3}

.
c) Let

X

be a Banach space and

Y

be a closed subspace of

X

. Let

π : X \to X / Y

be canonical quotient map. Show that

π

is open.
10. State giving reasons, if the following statement are true or false.
a) A closed map on a normed space need not be an open map.
b)

c_{00}

is a closed subspace of

ℓ^{\infty}

.
c) The dual of a finite dimensional space is finite dimensional.
d) If

T_{1}

and

T_{2}

are positive operators on a Hilbert space

H

, then

T_{1} + T_{2}

is a positive operator on

H

.
e) On a normed space

X

, the norm function

∥ \cdot ∥ : X \to C

is a linear map.

MMT-006 Sample Solution 2024

mmt-006-solved-assignment-2024-ss-020cab3d-1c01-486f-9bdf-7506d86b97ee

a) Let $X = {f \in C [0, 1] : f (0) = 0}$

Y = {g \in x : \int_{0}^{1} g (t) d t = 0}

Prove that

Y

is a proper subspace of

X

. Is

Y

a closed subspace of

X

? Justify your answer.

Answer:

To prove that

Y

is a proper subspace of

X

and determine whether

Y

is a closed subspace of

X

, we need to examine the definitions and properties of these sets within the context of functional analysis.

$X$ and $Y$ Defined

$X = {f \in C [0, 1] : f (0) = 0}$ is the set of all continuous functions on the interval $[0, 1]$ that vanish at $0$ .
$Y = {g \in X : \int_{0}^{1} g (t) d t = 0}$ is the set of all functions in $X$ whose integral over $[0, 1]$ is $0$ .

Proving $Y$ is a Proper Subspace of $X$

To show that

Y

is a proper subspace of

X

, we must verify that

Y

satisfies the following criteria for being a subspace:

Non-emptiness: $Y$ contains the zero function, which is the function $g (t) = 0$ for all $t \in [0, 1]$ . This function clearly belongs to $X$ and satisfies the integral condition, so $Y$ is non-empty.
Closed under addition: If $g_{1}, g_{2} \in Y$ , then $g_{1} + g_{2} \in Y$ . This is because the integral of the sum is the sum of the integrals, each of which is $0$ , so their sum is also $0$ .
Closed under scalar multiplication: If $g \in Y$ and $α$ is a scalar, then $α g \in Y$ . This follows because $\int_{0}^{1} α g (t) d t = α \int_{0}^{1} g (t) d t = α \cdot 0 = 0$ .

Since

Y

satisfies these criteria, it is a subspace of

X

. It is a proper subspace because there exist functions in

X

that do not satisfy the integral condition, such as

f (t) = t

, which is in

X

but not in

Y

since

\int_{0}^{1} t d t = 1 / 2 \neq 0

Is $Y$ a Closed Subspace of $X$ ?

A subspace

Y

is closed in

X

if it contains all its limit points; that is, if a sequence of functions

{g_{n}}

Y

converges uniformly to a function

g

, then

g

must also be in

Y

To show

Y

is closed, consider a sequence

{g_{n}} \subset Y

that converges uniformly to

g \in X

. We need to show that

g \in Y

, meaning

\int_{0}^{1} g (t) d t = 0

Uniform convergence of

{g_{n}}

g

implies that for every

ϵ > 0

, there exists an

N

such that for all

n \geq N

, we have

| g_{n} (t) - g (t) | < ϵ

for all

t \in [0, 1]

. By the properties of integrals and the limit of a sequence of functions,

lim_{n \to \infty} \int_{0}^{1} g_{n} (t) d t = \int_{0}^{1} lim_{n \to \infty} g_{n} (t) d t = \int_{0}^{1} g (t) d t .

Since each

g_{n} \in Y

, we have

\int_{0}^{1} g_{n} (t) d t = 0

. Thus, the limit of these integrals as

n \to \infty

is also

0

, which means

\int_{0}^{1} g (t) d t = 0

, and hence

g \in Y

Therefore,

Y

is a closed subspace of

X

because it contains all its limit points under uniform convergence.

b) Let

X = L^{p} [0, 1]

and

x = x (t) = t^{2}

. Find

∥ x ∥_{p}

for

p = 4

and

\infty

Answer:

To find the

L^{p}

norm of

x (t) = t^{2}

on the interval

[0, 1]

for

p = 4

and

p = \infty

, we’ll use the definitions of the

L^{p}

norms.

For $p = 4$

The

L^{p}

norm for

p = 4

is defined as

∥ x ∥_{4} = {(\int_{0}^{1} | t^{2} |^{4} d t)}^{1 / 4}

This simplifies to

∥ x ∥_{4} = {(\int_{0}^{1} t^{8} d t)}^{1 / 4}

To compute this integral, we use the formula for the integral of

t^{n}

, which is

\frac{t^{n + 1}}{n + 1}

for

n \neq - 1

, from

0

1

\int_{0}^{1} t^{8} d t = \frac{t^{9}}{9} |_{0}^{1} = \frac{1}{9}

Therefore,

∥ x ∥_{4} = {(\frac{1}{9})}^{1 / 4}

For $p = \infty$

The

L^{\infty}

norm, or the supremum norm, is defined as the essential supremum of

| x (t) |

over the interval. For

x (t) = t^{2}

[0, 1]

, this is simply the maximum value of

t^{2}

on the interval, which occurs at

t = 1

. Therefore,

∥ x ∥_{\infty} = max_{t \in [0, 1]} | t^{2} | = 1^{2} = 1

Calculation

Let’s calculate

∥ x ∥_{4}

explicitly:

∥ x ∥_{4} = {(\frac{1}{9})}^{1 / 4}

This calculation yields:

∥ x ∥_{4} = {(\frac{1}{9})}^{1 / 4} = \frac{1}{3^{1 / 2}} = \frac{1}{\sqrt{3}}

So, summarizing:

For $p = 4$ , $∥ x ∥_{4} = \frac{1}{\sqrt{3}}$ .
For $p = \infty$ , $∥ x ∥_{\infty} = 1$ .

c) Let E be a subset of a normed space

X, Y = span E

and

a \in X

. Show that

a \in \bar{Y}

if and only if

f (a) = 0

whenever

f \in X^{'}

and

f = 0

everywhere on

E

Answer:

To show that

a \in \bar{Y}

if and only if

f (a) = 0

for every

f \in X^{'}

(the dual space of

X

) where

f = 0

E

, we’ll use some fundamental properties of normed spaces, their duals, and the concept of closure.

Definitions:

$X$ is a normed space, and $X^{'}$ is its dual space, consisting of all continuous linear functionals on $X$ .
$E$ is a subset of $X$ , and $Y = span E$ is the subspace spanned by $E$ .
$\bar{Y}$ denotes the closure of $Y$ in $X$ , which includes all limits of convergent sequences in $Y$ .

( $\Rightarrow$ ) If $a \in \bar{Y}$ , then $f (a) = 0$ for every $f \in X^{'}$ where $f = 0$ on $E$ :

Assume

a \in \bar{Y}

. This means that for every

ϵ > 0

, there exists

y \in Y

such that

∥ a - y ∥ < ϵ

, since

a

can be approximated arbitrarily closely by elements of

Y

Let

f \in X^{'}

be such that

f = 0

E

. Since

Y = span E

f

also vanishes on

Y

because any linear combination of elements in

E

(which

f

maps to

0

) will also be mapped to

0

f

Given

a \in \bar{Y}

, and considering

f

is continuous, we have

f (a) = lim_{y \to a, y \in Y} f (y)

. Since

f (y) = 0

for all

y \in Y

, it follows that

f (a) = 0

( $\Leftarrow$ ) If $f (a) = 0$ for every $f \in X^{'}$ where $f = 0$ on $E$ , then $a \in \bar{Y}$ :

Assume

f (a) = 0

for every

f \in X^{'}

where

f = 0

E

. Suppose, for the sake of contradiction, that

a \notin \bar{Y}

. By the Hahn-Banach theorem, there exists a continuous linear functional

f \in X^{'}

such that

f (a) \neq 0

and

f = 0

\bar{Y}

(since

a

and

\bar{Y}

can be separated).

However, since

f = 0

\bar{Y}

, and

E \subseteq Y \subseteq \bar{Y}

, it follows that

f = 0

E

. But we assumed that

f (a) = 0

for all such

f

, contradicting the existence of such an

f

that separates

a

from

\bar{Y}

Therefore, our assumption that

a \notin \bar{Y}

must be false, implying

a \in \bar{Y}

Conclusion:

We’ve shown that

a \in \bar{Y}

if and only if

f (a) = 0

for every

f \in X^{'}

where

f = 0

E

, using the properties of normed spaces, their duals, and the Hahn-Banach theorem for the separation argument.

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IGNOU MMT-006 Solved Assignment 2024 | M.Sc. MACS

IGNOU MMT-006 Assignment Question Paper 2024

mmt-006-assignment-question-paper-aab71826-936e-4140-892f-f2b7ca3c6de5

MMT-006 Sample Solution 2024

mmt-006-solved-assignment-2024-ss-020cab3d-1c01-486f-9bdf-7506d86b97ee

$X$ and $Y$ Defined

Proving $Y$ is a Proper Subspace of $X$

Is $Y$ a Closed Subspace of $X$ ?

For $p = 4$

For $p = \infty$

Calculation

Definitions:

( $\Rightarrow$ ) If $a \in \bar{Y}$ , then $f (a) = 0$ for every $f \in X^{'}$ where $f = 0$ on $E$ :

( $\Leftarrow$ ) If $f (a) = 0$ for every $f \in X^{'}$ where $f = 0$ on $E$ , then $a \in \bar{Y}$ :

Conclusion:

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IGNOU MMT-006 Solved Assignment 2024 | M.Sc. MACS

IGNOU MMT-006 Assignment Question Paper 2024

mmt-006-assignment-question-paper-aab71826-936e-4140-892f-f2b7ca3c6de5

MMT-006 Sample Solution 2024

mmt-006-solved-assignment-2024-ss-020cab3d-1c01-486f-9bdf-7506d86b97ee

X X XXX and Y Y YYY Defined

Proving Y Y YYY is a Proper Subspace of X X XXX

Is Y Y YYY a Closed Subspace of X X XXX?

For p = 4 p = 4 p=4p=4p=4

For p = ∞ p = ∞ p=oop=\inftyp=∞

Calculation

Definitions:

( ⇒ ⇒ =>\Rightarrow⇒) If a ∈ Y ¯ a ∈ Y ¯ a in bar(Y)a \in \bar{Y}a∈Y¯, then f ( a ) = 0 f ( a ) = 0 f(a)=0f(a) = 0f(a)=0 for every f ∈ X ′ f ∈ X ′ f inX^(‘)f \in X’f∈X′ where f = 0 f = 0 f=0f = 0f=0 on E E EEE:

( ⇐ ⇐ lArr\Leftarrow⇐) If f ( a ) = 0 f ( a ) = 0 f(a)=0f(a) = 0f(a)=0 for every f ∈ X ′ f ∈ X ′ f inX^(‘)f \in X’f∈X′ where f = 0 f = 0 f=0f = 0f=0 on E E EEE, then a ∈ Y ¯ a ∈ Y ¯ a in bar(Y)a \in \bar{Y}a∈Y¯:

Conclusion:

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$X$ and $Y$ Defined

Proving $Y$ is a Proper Subspace of $X$

Is $Y$ a Closed Subspace of $X$ ?

For $p = 4$

For $p = \infty$

( $\Rightarrow$ ) If $a \in \bar{Y}$ , then $f (a) = 0$ for every $f \in X^{'}$ where $f = 0$ on $E$ :

( $\Leftarrow$ ) If $f (a) = 0$ for every $f \in X^{'}$ where $f = 0$ on $E$ , then $a \in \bar{Y}$ :