Solve your IGNOU Doubts

Question:

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 of X such that for every $x \in D, \left \{ F(x):F \in f \right \}$ 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 $C^1[0,1]$ of all C¹ functions on [0,1] endowed with the uniform norm induced from the space C[0,1] and consider the differential operator $D:(C^1[0,1],\mid \mid .\mid \mid_\infty) \to (C[0,1],\mid \mid .\mid \mid_\infty)$ 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 c₀₀ For $x = (x_1,x_2,....,x_n,....) \in c_{00}$ 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 $\mid \mid x\mid \mid = sup\left | x_n \right |$

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Question:

Let E be a subset of a normed space X, Y = span E and $a \in X$ Show that $a \in \overline{Y}$ if and only if f(a) = 0 whenever $f \in X'$ and f = 0 everywhere on E

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Question:

Let $X = L^P[0,1]$ and $x =x(t) = t^2$ Find $\mid x\mid _P$ for P = 4 and $\infty$ .

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Question:

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

$Y = \begin{Bmatrix} g \in x: \int_{0}^{1}g(t)dt = 0 \end{Bmatrix}$

Prove that Y is a proper subspace of X. Is Y a closed subspace of X? Justify your answer

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Question:

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

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Question:

Solve the following set of congruences:

$x \equiv 2 (mod \: 17)$

$3x \equiv 4 (mod \: 19)$

$x \equiv 7 (mod \: 23)$

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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:

In this exercise, we ask you to find the Sylow p-subgroups of the dihedral group $D_n = {x,y :x^n,y^2,yxyx}, n \in N, n\geq 2$

(a) Let p be an odd prime that divides n, $n = p^rl, p \nmid l$ . Suppose $C = \langle x^l \rangle$ . Show that C is the unique Sylow p-subgroup of D_n .

(b) Prove the relation $y^ix^jy^kx^l = \left\{\begin{matrix} y^ix^{j+1} & if\: k\: is \: \: even \\ y^{i+k}x^{l-j} & if\: k\: is \: \: odd \end{matrix}\right.$

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

(d) Suppose n is even, n = 2^km, where $2 \nmid m, k\geq 2$ . Let $N = \langle x^m \rangle$ and $H = \langle y \rangle$ . Show that HN is a subgroup of D_n . What is its order?

(e) 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:

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

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Question:

Show that, to prove that SP₂(R) acts transitively on GL₂(R), it is enough to show that, for any vector $\begin{bmatrix} a\\ b \end{bmatrix} \neq 0 \in R^2$ , there is a $2\times 2$ symplectic matrix with $\begin{bmatrix} a \\ b \end{bmatrix}$ as the first column. (Hint: For any matrix A, what is $A\begin{bmatrix} I \\ 0 \end{bmatrix}$ ?)

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Question:

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

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Question:

The aim of this exercise is to show that SP₂(R)acts transitively on R² \ {0}.

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Question:

Suppose that $m = \begin{bmatrix} A & B\\ C & D \end{bmatrix}$ is $2n\times 2n$ matrix where A,B,C and D are $n\times n$ matrices. Show that M is symplectic if and only if the following conditions are satisfied:

$A^tD - C^tB = I$

$A^tC - C^tA = 0$

$B^tD - D^tB = 0$

(Hint: Use block matrix multiplication.) Also, check that the matrix $B^tD - D^tB = 0\begin{bmatrix} 0 & -A\\ A& 0 \end{bmatrix}$ , where A is a $n\times n$ orthogonal matrix, is a symplectic matrix.

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Question:

If F is a finite field show that there is always an irreducible polynomial of the form $x^3-x+a$ where $a \in F$ .(Hint: Show that $x \mapsto x^3 - x$ is not a surjective map.)

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Question:

By looking at the factorisation of $x^9 - x \in F_3[x]$ guess the number of irreducible polynomials of degree 2 over $F_3$ . Find all the irreducible polynomials of degree 2 over $F_3$ .

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Question:

If $char(F) \neq 2$ , show that a polynomial $ax^2 + bx + c$ is irreducible iff $b^2-4ac \notin F^{*2}$ where $F^{*2}$ is the group of squares in $F^{*}$ .