Solve your IGNOU Doubts

Question:

Show that a set A in a metric space is closed if and only if every convergent sequence in A converges to a point of A.

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Question:

If E is a subset of $\mathbb{R}$ with standard metric, then show that $(\bar{E})^c = (E^c)^\circ$ .

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Question:

Which of the following subsets of $\mathbb{R}$ are compact w.r.t. the metric given against them. Justify your answer.
i) $A = (0, 1)$ in $\mathbb{R}$ of $-\mathbb{R}$ with standard metric.
ii) $A = [3, 4] - \mathbb{R}$ with discrete metric.
iii) $\{(x, y) \in \mathbb{R}^2 \quad y > 0\} - \mathbb{R}^2$ with standard metric.

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Question:

Prove that if an open set U can be written as the union of pariwise disjoint family V of open connected subsets, then these subsets must be the components of U. Use this theorem to find the components of the set $D \cup E$ where

$$D = \{(x, y) \in \mathbb{R}^2 : \|(x+1, y)\| = 1\} \text{ and}$

$$E = \{(x, y) \in \mathbb{R}^3 : \|x-1, y\| = 1\}.$

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Question:

Find the Fourier series of $f(t) = t^2$ on $[-\pi, \pi]$ .

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Question:

For the equation $x^2 + y^2 + z^3 = 0$ , at which points on its solution set, can we assured that there is a neighbourhood of the point in which the surface given by the equation can be described by an equation of the form $z = f(x, y)$ .

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Question:

Let $f : \mathbb{R}^3 \setminus \{(0,0,0)\} \to \mathbb{R}^3 \setminus \{(0,0,0)\}$ be given by

$$f(x, y, z) = \left( \frac{x}{x^2 + y^2 + z^2}, \frac{y}{x^2 + y^2 + z^2}, \frac{z}{x^2 + y^2 + z^2} \right)$

Show that f is locally invertible at all points in $\mathbb{R}^3 \setminus \{(0,0,0)\}$ .

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Question:

Find and classify the extreme values of $(x, y) = xy$

Subject to the constraint

$$\frac{x^2}{8} + \frac{y^2}{2} - 1 = 0.$

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Question:

Find the Lebesgue integral of the function f given by

$$f = \chi_{[0, 1]} + 2\chi_{(3, 5]} + \chi_{[6, 9]}.$

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Question:

Obtain the second Taylor’s series expansion for the function given by

$$f(x, y) = xy^2 + 5xe^y \text{ at } \left(1, \frac{\pi}{2}\right).$

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Question:

Does Cantor’s intersection theorem hold for the metric space $X = (0, 1]$ with the standard metric? Justify your answer.

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Question:

Consider $f : \mathbb{R}^3 \to \mathbb{R}^3$ given by

$$f(x, y, z) = (2x + 3y + z, xy, yz, xz)$

Find f(2, 0, -1).

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Question:

Find the interior, closure, the set of limit points and the boundary of the set

$$A = \{(x, y) \in \mathbb{R}^2 : y = 1\}$

in $\mathbb{R}^2$ with the standard metric.

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Question:

Find the interior, closure, the set of limit points and the boundary of the set

$$A = \{(x, y) \in \mathbb{R}^2 : y = 1\}$

in $\mathbb{R}^2$ with the standard metric.

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Question:

1. State whether the following statements are true or false. Justify your answers. $5 \times 2 = 10$
a) The outer measure m^ of the set $A = \{x \in \mathbb{R} : x^2 = 1\} \cup [-3, 2]$ is 0.
b) A finite subset of a metric space is totally bounded.
c) A connected subspace in a metric space which in not properly contained in any other connected subspace is always open.
d) The surface given by the equation $x + y + z - \sin(xyz) = 0$ can also be described by an equation of the form $z = f(x, y)$ in a neighbourhood of the point (0, 0).
e) A real valued function f on [a, b] is continuous if it is integrable on [a, b].

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Question:

Prove the following result:

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?