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Submit Papers đŠNote :- All questions from each section carry equal marks. All questions are compulsory and answer limit are approximately 250 words. Start the answer of each section from new page. Maximum limit of pages of answer booklet are approximately 16 pages. Answer should be written by the student in his/her own handwriting mandatory. The first page of answersheet should be download by the student from university website is mandatory.
Let H1, H2 be Hilbert spaces, S:H1 → H2 and T:H1 → H2 bounded linear operators and x any scalar. Then prove that
(a) <T*y x>=<y, Tx>
: x∈ H1
y∈ H2
(b) (5+T)*=5*+T*
Define Partially ordered set, power set and give statement of Zorn's lemma.
Prove that every finite dimensional normed space is reflexive.
Let (Xn) be a weakly convergent sequence in a normed space x, say Xn &xrightarrow;w x. Then prove that
(a) The weak limit x of (Xn) is unique.
(b) The sequence ( (||Xn||) is bounded.
Explain open mapping and open unit ball.