Solve your IGNOU Doubts

Question:

Define the spectral radius of a bounded linear operator $A \in BL(X)$ .Find the spectral radius of A in $BL(R^3)$ ,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 $R^3$ .

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Question:

Let {A_n} be a sequence of unitary operators in BL(H). Prove that if A $||A_n - A|| \to 0, A \in BL(H)$ then A is unitary.

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Question:

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.

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Question:

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).

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Question:

Let X be a normed space and Y be proper subspace of X. Show that the interior Y⁰ of Y is empty

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Question:

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.

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Question:

Let $X = c_{00}$ with $||.||_p$ Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example

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Question:

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

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Question:

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.

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Question:

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

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Question:

Let X be an inner product space and $x,y \in X$ Prove that $x \perp y$ if and only if $||kx + y||^2 = ||kx||^2 + ||y^2||, k \in K$

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Question:

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.

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Question:

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

i) $T : \mathbb{R}^3 \to \mathbb{R}^2$ 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)

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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 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