Solve your IGNOU Doubts

Question:

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.

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Question:

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

i) $T:\mathbb{R}^3\to\mathbb{R}^2\text{given by}T(x,y,z)=(x,z).$

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

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Question:

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.

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Question:

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.

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Question:

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.]

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Question:

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 $D:(C^1[0,1],\|\cdot\|_\infty)\to(C[0,1],\|\cdot\|_\infty)$ 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.

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Question:

Consider the space $c_{00}.\text{For}x=(x_1,x_2,\dots,x_n,\dots)\in c_{00}$ , 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 $\|x\|=\sup_n|x_n|$

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Question:

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

a) The function $\|-\|$ defined on $\mathbb{R}_{n}$ as:

$\|x\|=\sum_{j=1}^{n}|a_j|\quad\text{for}x=(a_1,a_2,\dots,a_n)\in\mathbb{R}^n$

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'.

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Question:

Show that $\mathbb{Q}(\sqrt{-19})$ is not a UFD by giving two different factorisations of 20.

Question no.	Block 1	Block 2	Block 3	Block 4	Block 5
2 a)	5
2 b)	3
2 c)		2
3 c)			10
4 a)	4
4 b)	3
4 c)			3
5 a)				2
5 b)				5
5 c)				3
6 a)				2
6 b)				6
6 c)				2
7 a)		4
7 c)		1
7 c)		3
7 c)		2
8 a)	3
8 b)	4
8 c)	2
8 d)	3
8 e)	3
9 a)		5
9 b)			5
9 c)			5
Total	30	17	23	20	0

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Question:

Solve the following set of congruences:

$x\equiv 2\pmod{17}$

$3x\equiv 4\pmod{19}$

$x\equiv 7\pmod{23}$

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Question:

Let $G=\langle a,b\mid a^2,b^3,aba^{-1}b^{-1}\rangle$ . Show that G is the cyclic group of order six.

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Question:

Suppose n is as in the previous part. Find all the Sylow 2-supgroups of D_n. Describe them in terms of x and y.

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Question:

Suppose n is even, n = 2^km, where 2 + m, k ≥ 2. Let N = (x^m) and H = (y). Show that H N is a subgroup of D_n. What is its order?

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Question:

Find all the Sylow 2-subgroups of D_n, when n is odd. Describe them in terms of x and y.

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Question:

Prove the relation

$y^j x^k y^l=\begin{cases}y^j x^{j+l}&\text{if k is even}\\y^{j+k}x^l&\text{if k is odd}\end{cases}$

Further, find all the elements of order 2 in D_n.

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Question:

Let p be an odd prime that divides n, n = p^r l, p + 1. Suppose C = (x¹). Show that C is the unique Sylow p-subgroup of D_n.

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Question:

In this exercise, we ask you to find the Sylow ?-subgroups of the dihedral group

$D_n=\langle x,y:x^n,y^2,yxyx\rangle,\quad n\in\mathbb{N},n\ge 2.$

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Question:

symplectic matrix with $\begin{bmatrix}a\\b\end{bmatrix}$ as the first column . (Hint: For any matrix A, what is $A\,\,\frac{1}{2}$ ?)

(iii) Complete the proof by showing that, given any non-zero $\begin{bmatrix}a\\b\end{bmatrix}$ vector non-zero vector, there is always a $\begin{bmatrix}a'\\b'\end{bmatrix}$ such that $\begin{bmatrix}a&a'\\b&b'\end{bmatrix}$ is symplectic.

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Question:

Show that

(i) Show that a matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}\in GL_2(\mathbb{R})$ is symplectic if and only if ad - bc = 1.

(ii) Show that, to prove that SP₂ $\left(\mathbb{R}\right)$ acts transitively on GL₂ $\left(\mathbb{R}\right)$ , it is enough to show that, for any vector $\begin{bmatrix}a\\b\end{bmatrix}\ne\mathbf{0}\in\mathbb{R}^2$ , there is a 2 x