IGNOU MMT 6 SOLVED ASSIGNMENT

Ques 1.

State whether the following statements True or False? Justify your answers: 
a) The function  defined on  as:
 for 
is a norm.

Ques 2.

) C₀ is a Banach space.

Ques 3.

State whether the following statements True or False? Justify your answers: 
a) The function  defined on  as:
 for 
is a norm.
b) C₀ 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'.

Ques 4.

Consider the space c₀₀. For , define . Show that f is a linear functional which is not continuous w.r.t the norm .

Ques 5.

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  defined by . 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.

Ques 6.

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.]

Ques 7.

) Let X be a Banach space, Y be a normed linear space and  be a subset of B(X, Y). If  is not uniformly bounded, then there exists a dense subset D of X such that for every  is not bounded in Y.

Ques 8.

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.

Ques 9.

) Which of the following maps are open? Give reasons for your answer.
i)  given by .
ii)  given by .

Ques 10.

 Let  be given by . Show that f is continuous w.r.t the supnorm and f is not continuous w.r.t the p-norm.

Ques 11.

Let X be an inner product space and . Prove that  if and only if .

Ques 12.

et  and F be the set of all  in H such that . Find . Verify that every  can be expressed as  where  and 

Ques 13.

Given an example of an Hilbert space H and an operator A on H such that  is empty. Justify your choice of example. 

Ques 14.

Let A be a normal operator on a Hilbert space X. Show that  where  denotes the approximate eigen spectrum of A and  denotes the spectrum of A. 

Ques 15.

) Let  with . Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example. 

Ques 16.

Give one example of each of the following. Also justify your choice of example. 
i) A self-adjoint operator on .
ii) A normal operator on a Hilbert space which is not unitary.

Ques 17.

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

Ques 18.

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). 

Ques 19.

Define a Hilbert-Schmidt operator on a Hilbert space H and give an example. Is every Hilbert-schmidt operator a compact operator? Justify your answer. 

Ques 20.

Let  be a sequence of unitary operators in BL(H). Prove that if , then A is unitary.

Ques 21.

 Define the spectral radius of a bounded linear operator . Find the spectral radius of A in , where A is given by the matrix

with respect to the standard basis of .

Ques 22.

Let X be a Banach space and Y be a closed subspace of X. Let  be canonical quotient map. Show that  is open.

Ques 23.

Give an example of a compact linear map on l².

Ques 24.

Give an example of a positive operator on 

Ques 25.

Prove the following result:

Ques 26.

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  of a non-zero real numbers with  and an orthonormal set  in H such that

Further, mention in which step of the proof it is used that A is a compact self-adjoint operator. Explain why?

Ques 27.

State whether the following statements True or False? Justify your answers: 
a) The function  defined on  as:
 for 
is a norm.

Ques 28.

) C₀ is a Banach space.

Ques 29.

State whether the following statements True or False? Justify your answers: 
a) The function  defined on  as:
 for 
is a norm.
b) C₀ 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'.

Ques 30.

Consider the space c₀₀. For , define . Show that f is a linear functional which is not continuous w.r.t the norm .

Ques 31.

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  defined by . 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.

Ques 32.

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.]

Ques 33.

) Let X be a Banach space, Y be a normed linear space and  be a subset of B(X, Y). If  is not uniformly bounded, then there exists a dense subset D of X such that for every  is not bounded in Y.

Ques 34.

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.

Ques 35.

) Which of the following maps are open? Give reasons for your answer.
i)  given by .
ii)  given by .

Ques 36.

 Let  be given by . Show that f is continuous w.r.t the supnorm and f is not continuous w.r.t the p-norm.

Ques 37.

Let X be an inner product space and . Prove that  if and only if .

Ques 38.

et  and F be the set of all  in H such that . Find . Verify that every  can be expressed as  where  and 

Ques 39.

Given an example of an Hilbert space H and an operator A on H such that  is empty. Justify your choice of example. 

Ques 40.

Let A be a normal operator on a Hilbert space X. Show that  where  denotes the approximate eigen spectrum of A and  denotes the spectrum of A. 

Ques 41.

) Let  with . Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example. 

Ques 42.

Give one example of each of the following. Also justify your choice of example. 
i) A self-adjoint operator on .
ii) A normal operator on a Hilbert space which is not unitary.

Ques 43.

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

Ques 44.

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). 

Ques 45.

Define a Hilbert-Schmidt operator on a Hilbert space H and give an example. Is every Hilbert-schmidt operator a compact operator? Justify your answer. 

Ques 46.

Let  be a sequence of unitary operators in BL(H). Prove that if , then A is unitary.

Ques 47.

 Define the spectral radius of a bounded linear operator . Find the spectral radius of A in , where A is given by the matrix

with respect to the standard basis of .

Ques 48.

Let X be a Banach space and Y be a closed subspace of X. Let  be canonical quotient map. Show that  is open.

Ques 49.

Give an example of a compact linear map on l².

Ques 50.

Give an example of a positive operator on 

Ques 51.

Prove the following result:

Ques 52.

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  of a non-zero real numbers with  and an orthonormal set  in H such that

Further, mention in which step of the proof it is used that A is a compact self-adjoint operator. Explain why?

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