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Submit Papers đŠTotal No. of Questions : 11
Total No. of Printed Pages : 6
(i)
EF-457
M.A./M.Sc. IIIrd Semester (Reg./Pvt./ATKT)
Examination, 2021-22
Maths
Functional Analysis-I
Time: 3 Hours
[Maximum Marks : Reg.= 85, Pvt.= 100]
Note:- Attempt all questions.
SECTION - 'A'
Objective Type Questions
1. Choose the correct answer :
(i) A metric space (X, d) with natural metric is
(a) Normed linear space
(b) Banach space
(c) Hilbert space
(d) None of these
(ii) If in an inner product space x, x + y then.
(a) ||x + y||² = ||x||² - ||y||²
(b) ||x + y||² = ||x||²
(c) ||x + y||² = ||x||² + ||y||²
(d) None of these
(iii) All norms are equivalent if the dimension of normed linear space is.
(a) Infinite
(b) finite
(c) Not defined
(d) None of these
(iv) If M is an m - dimensional sub space of an n-dimensional linear space N, then the dimension of the quotient space N/M is
(a) n
(b) m-n
(c) n-m
(d) None of these
(v) T: X → Y is a bounded linear operator then T-1: R(T) → X is
(a) Bounded
(b) Closed
(c) Need not be bounded
(d) None of these
SECTION - 'B'
Short Answer Type Questions
2. Define Banach space and give an example which is not Banach space.
OR
Prove that in a normed linear space every convergent sequence is a cauchy sequence.
3. Prove that every finite dimensional subspace Y of a normed linear space X is closed is X.
OR
A compact subsect M of a mertric space is closed and bounded.
4. Let T be a linear operator then prove that the range R (T) and null space N (T) is a vector space.
OR
Let X, Y be vector spaces, both real or both complex. Let T: D(T) → Y be a linear operator with domain D (T) ⊂ X and range R (T) ⊂ Y. Then if T-1 exists, it is a linear operator.
5. Prove that a finite dimensional vector space is algebraically reflexive.
OR
Prove that the dual space of R* is R*.
6. Prove that every Hilbert space is reflexive.
OR
Let M be a complete subspace Y and x ∈ X fixed. then prove that z= x − y is orthogonal to Y.
SECTION - 'C'
Long Answer Type Questions
7. Let {xâ, ......., xn} be a linearly in d e p e n d e n t set of vectors in a normed space X. Then there is a number C > 0 such that for every choice of scalars αâ, ......., αn Prove that
||αâxâ + αâxâ + ....... + αnxn|| ≥ C (||αâ| + ....... + |αn||) : : C > 0
OR
Prove that every finite dimensional normed space is complete.
8. State and prove Riesz's lemma.
OR
Let M be a closed linear subs space of a normed linear space x.
The quotient space X/M is a normed linear space with the norm
||x + M|| = inf {||x + m|| : m ∈ M}
9. If a normal space X is finite dimensional, then prove that every operator on X is bounded.
OR
Let T: D(T) → Y be a bounded linear operator, where D(T) lies in a normed space X and Y is a Banach space. Then T has an extension ∼T : D(∼T) → Y Where ∼T is a bounded linear operator of norm ||∼T|| = ||T||.
10. If y is a Banach space, then prove that B (X, Y) is a Banach space.
OR
Prove that the dual space of lp is lc.
11. Let Y be any closed subspace of a Hilbert space H. then H = Y ⊕ Z. Z = Y⊥
OR
State and prove Riesz's theorem.