# Dr.A.P.J.Abdul Kalam University MSC MATHS IV SEM JUNE20 Functional Analysis-II [MSM401T] Aug-Sep 2020 Question Paper

Dr.A.P.J.Abdul Kalam University Old Question Papers

Master of Science (Mathematics)

Fourth Semester Main Examination, Aug-Sep 2020

Functional Analysis-II [MSM401T]

Time: 3:00 Hrs Max Marks 85

Note : Attempt all questions. Question no. 1 to Q. no.5 has two parts. Part A is 10 marks and part

B is 7 marks.

Q.1 (a) Show that the norm of an isometry is one.

(b) Explain Bessel inequality.

OR

(a) Explain Inner Product Space.

(b) Define orthonormal sets.

Q.2 (a) Define total orthonormal. Let be a subset of an inner product space X, which is total in

X. then prove that

(b) State and prove Riesz’s theorem.

OR

(a) State and prove Riesz representation theorem.

(b) Let H be a separable Hilbert space then prove that every orthonormal set

in H is countable.

Q.3 (a) Explain Hilbert adjoint operators.

(b) If H is complex and is real for all the operator T is self-adjoint. Prove it T:H

H be a bonded linear operator on a Hilbert space H.

OR

(a) Explain normal operator.

(b) Show that the unitary operators on a Hilbert space H from a group.

Q.4 (a) State and Prove Uniform Boundedness theorem.

(b) Explain weak convergence.

OR

(a) Explain strong convergence.

(b) If a normed space X is reflexive, show that X’ is reflexive.

Q.5 (a) State and Prove Closed graph theorem.

(b) Show that an open mapping need not map closed sets onto closed sets.

OR

(a) Prove that a bounded linear operator. T from a Banach space X onto a Banach space y has the

property that the image (

) of the open unit ball Bo= B(0,1) X contains an open ball about .

(b) Define closed linear operators.

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