Solve your IGNOU Doubts

Question:

Let ]1,0[ X = C′ and ] Y = 11,0[C and let T : X → Y be the linear operator from X to Y given by ,f )f(T = ′ the derivative of f on ].1,0[ Show that T is not continuous.

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Question:

Let H be a Hilbert space. For any subset A of ,H define . A ⊥ If ,H A ⊆ B ⊆ then show that:

i) B¹⊆ A

ii) A ⊆ A ^⊥⊥

State conditions on A^⊥⊥ =A

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Question:

Let X be a vector space. Let $\left \| . \right \|^{1}and \left \| . \right \|^{2}$ be two norms on .X When are these norms said to be equivalent? Justify your answer.

Let . X = $\mathbb{R}^{3}$ For x=(x_1,x₂,x₃)

Let $\left \| x \right \|=\left \| x \right \|+\left \| x_{2} \right \|+\left \| x_{3} \right \|$

$\left \| x \right \|^{2}=\sqrt{x_{1}+x_{2}+x_{3}^{2}}$

Show that $\left \| . \right \|^{1}$ and $\left \| . \right \|^{1}$ are equivalent.

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Question:

Let X be an inner product space with the inner product given by <, > . For ,X x ∈ define the function $\left \| . \right \|:X\rightarrow K$ given by $\left \| x \right \|=< x,< -x> 1/2$ the non negative square root of < x, x > . Show that $\left \| . \right \|:X\rightarrow K$ defines a norm on X and $|< (x,y)> |\leq \left \| x \right \|\left \| y \right \|$ for all .X ,x y∈ Also show that for all ,X , $\left \| x+y \right \|^{2} +\left \| x-y \right \|^{2}=2\left \| x \right \|^{2}+\left \| y \right \|^{2}$ .

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Question:

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Question:

Define Eigen Spectrum of a bounded linear operator on a Banach space. Show that the eigen spectrum of the operator T on l² given by ) (T , $\alpha _{1},\alpha _{2}$ ......... ) = $(0,\alpha _{1},\alpha _{2}.....)$ is empty .

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Question:

Let T be a linear map defined on X by $T(f)=f\frac{1}{2}$

Show that T is a bounded linear map such that T = .1

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Question:

Let ]1,0[C X = with Sup norm defined by f = Sup{| )x(f |}. ]1

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Question:

Let } u{ n be the sequence in l² with 1 in the th n place and zeroes else where prove that the set } u{ n is an orthonormal basis for l ²

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Question:

Let A be an operator on a Hilbert space .H Show if Ax A x ∗ = for every ,H x ∈ then A is normal. Is it converse true? Justify

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Question:

Show that Q defined on $(C\begin{bmatrix} 0,1\left \| \right \| \end{bmatrix})$ by $Q(x)=\int tx(t)$ dt)t(xt is a bounded linear functional. Calculate $\mathbb{Q}$ .

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Question:

If H is a Hilbert space and SCH, show that . S S ⊥ ⊥⊥⊥ = When S is the same as ? S ⊥⊥ Justify.

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Question:

State the principle of uniform boundedness. Use it to show that a set E in a normed space X is bounded if )E(f is bounded in K for every f ∈ X′.

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Question:

Are Hahn-Banach extensions always unique? Justify.

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Question:

In a Hilbert space. Prove that x _n → x provided $\left \| X_{a} \right \|\left \| \rightarrow \right \|x\left \| \right \|and (X_{n,X})\rightarrow (x,x)$

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Question:

Show how a real linear functional u on a complex linear normed space gives rise to a complex linear functional .f What is the relation between the boundedness of u and that of f ?

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Question:

Check whether a finite dimensional normed linear space is reflexive? Justify your answer.

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Question:

Show that the map 3 T : R³→ R² given by ) T(x₁ , x ₂, x₃ ) =(x1,+x2,+x3 is an open map

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Question:

Characterise all bounded linear functionals on a Hilbert space.

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Question:

If A is a bounded linear operator on a Hilbert space such that , AA^* = I then A A^*= I.