MAMT06 Analysis and Advanced Calculus M.Sc Question Bank : vmou.ac.in
Name of the University : Vardhman Mahaveer Open University
Degree : M.Sc
Department : Mathematics
Subject Code/Name : MAMT-06 – Analysis and Advanced Calculus
Year : II
Document Type : Question Bank
Website : vmou.ac.in
Download Model/Sample Question Paper : https://www.pdfquestion.in/uploads/vmou.ac.in/3592.-MAMT-06_74.pdf
VMOU Analysis & Advanced Calculus Question Paper
1. Define Norm and write the set of axions of Normal linear space.
2. Write the Summability for a series S –
3. If N be a Normed linear space and x, y, – – then Prove
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4. Show that every normed linear space is a metric space.
5. If N be a normed linear space with the norm – . – , then prove that mapping
6. Show that every convergent sequence in a normal linear space.
7. Show that the limit of a convergent sequence is unique.
8. Write the Reflexive, Symmetric and Transitive relations for factor –
9. Show that the linear spaces R(real) and (Complex) are normed linear spaces
10. If T be a linear transformation of a normed linear space N –
11. If T be a linear transformation from a normed –
12. If M be a closed linear subspace of a normed linear space N and T –
13. Show that the weak limit of a sequence is unique.
14. Show that on a finite dimensional linear space X, all norms are equivalent.
15. Prove that every compact subset of a normal linear space is complete.
16. Prove that every compact subset of a normal space –
17. Let n be a normed linear space and suppose the set –
18. Let B and – be Banach spaces. If T is a continuous linear transformation-
19. Let N be a real normed linear space and suppose –
20. If M be a closed linear subspace of a normed linear space N and –
21. Show that the space –
22. Show that the linear space –
23. Show that the inner product in a Hilbert space is jointly continuous
24. If x and y are any two vectors in a Hilbert space H, then prove
25. If x, y are any two vectors in a Hilbert space H, then prove that
26. Write the Pythagorean theorem statement and proof also.
27. If S is a non empty subset of a Hilbert space H.
28. Let M be a linear subspace of Hilbert space H.
29. If M is a closed linear subspace of a Hilbert space H.
30. If {- } is an orthonormal set in a Hilbert space H and if x is any vector in H,
31. Show that an orthonormal set S in a Hilbert space H is complete iff
32. Show that in the Hilbert space –
33. If an operator T on H is self-adjoint, then show that
34. If T be a self-adjoint operator, then show that –
35. If T is an arbitrary operator on Hilbert space H, then prove that T=0
36. Show that an operator T on a complex Hilbert space H is self=adjoint
37. Let A be the set of all self-adjoint operators in analytical maths vectotr T
38. Write the conditions so that :(a) The identity operator I and the zero operator.
39. Prove that an operator T on a Hilbert space H is normal iff
40. If T is a Normal Operator on H, then prove that –
42. If P is a projection on a Hilbert space H, then prove that :
43. If P is a projection on a Hilbert space H, then prove that :
44. Show that a closd linear subspace M of a Hilbert space H
45. Show that a closed linear subspace M of a Hilbert space H r
46. If P and Q are projections on closed linear subspace M and N
47. If x is an eigenvector of T corresponding to eigenvalue – ,
48. If x is an eigenvector of T, then show that x cannot corresponding T-
49. If T is a normal operator on a Hilbert space H,
50. If T is a normal operator on a Hilbert space H then prove