Download Original PDF
Get the official Barkatullah University print version scanned document.
🤝 Help Your Juniors!
Have previous year question papers that aren't on our website? Help the next batch of students by sending them to us! With your consent, we will proudly feature your name as a Top Contributor on our platform.
Submit Papers 📩Total No. of Questions : 11
Roll No.
Total No. of Printed Pages : 8
M.A./M.Sc. IIIrd Semester (Reg./Pvt./ATKT)
Examination, 2023-24
Maths
Functional Analysis-I
Examination, 2023-24
Maths
Functional Analysis-I
Time : 3 Hours]
[Maximum Marks :
Reg.=85
Pvt.=100
Reg.=85
Pvt.=100
Note :- Attempt all the questions from Section "A", "B" and "C". Internal choice is given where necessary. Marks are indicated.
SECTION - 'A'
Objective Type Questions
1x10=10
Objective Type Questions
1x10=10
1.
Choose the correct answer :
- Identity operator
- None of these
(i)
Any subset M of a normed space x is compact if and only if M is closed and bounded then :
- x is of finite dimensional
- x is of infinite dimensional
- x is of finite or infinite dimensional
- None of these
(ii)
If y be a closed linear subspace of a normed space x. Then a mapping T : X → X/Y defined by T(x) = x + Y is
- Linear transformation
- Continuous linear transformation
- Bounded linear transformation
- All of above .
(iii)
- Null space N(T) is closed but range R(T) is need not be closed
- Range R(T) is closed but null space N(T) is need not be closed
- Neither null space N(T) is closed nor range R(T) is closed
(iv)
If T:X → Y is bounded linear operator then T-1 : R(T) → X is
- Bounded
- Closed
- Need not be bounded
- None of these
(v)
The dual space x' of a normed space X is a :
- Normed space
- Linear space
- Banach space
- Hilbert space
(vi)
Which of the following is not a bounded operator :
- Differential operator
- Zero operator
(vii)
In any inner product space x, if x ∈ Y then the equation ||x + y||² + ||x − y||² = 2 (||x||² + ||y||²) is known as:
- Triangle inequality
- Law of parallelogram
- Schwarz inequality
- None of these
(viii)
If in the space Rⁿ, x, y ∈ Rⁿ where x = (p₁, p₂,..., pn) y = (η₁, η₂,..., ηn) and inner product defined as <x, y> = ∑i=1ⁿ pᵢηᵢ then Rⁿ is a
- Topological space
- Hausdorff space
- Hilbert space
- Banach space
(ix)
If f is a bounded linear functional on a complex normed space, then conjugate of f, f one is:
- Bounded
- Unbounded
- Linearly bounded
- None of these
(x)
If in an inner product space X, x,L Y then
- ||x+y||² - ||x||²
- ||x+y||² = ||x-y||²
- ||x+y||² = ||x||² + ||y||²
- None of these
SECTION - 'B'
Short Answer Type Questions
5x5=25
Short Answer Type Questions
5x5=25
2.
Define Banach space and give an example which is not Banach space.
OR
Prove that in normed space x if xn → x and yn → y implies xn + yn → x + y also αn → α and xn → x implies αn xn → αx.
3.
Define the equivalent norm and prove that on a finite dimensional vector space X any norm ||.|| is equivalent to any another norm ||.||0.
OR
If a normed space x has property that the closed unit ball
4.
M={x:||x|| ≤ 1} is compact then prove that x is finite dimensional.
5.
Let x and y be metric spaces and T : x → y a continuous mapping. Then prove that the image of a compact subset M of x under T is compact.
OR
Let T be a linear operator, then prove that the range R(T) is a vector space.
6.
Let x be an n-dimensional vector space and E = {e1,..., en} a basis for x. Then F = {f1,...,fn} given by
fi(ej) = δij = { 0 if j ≠ k
1 if j = k }
is a basis for the algebraic dual x* of x and dim x* = dim x = n.
1 if j = k }
OR
Prove that the dual space Rn is Rn.
SECTION - 'C'
Long Answer Type Questions
5x10=50
Long Answer Type Questions
5x10=50
7.
Show that a subspace Y of a Banach space x is complete if and only if the set Y is closed in x.
OR
Prove that every finite dimensional normed space in complete.
8.
Prove that in a finite dimensional normed space X, any subset MCX is compact if and only if M is closed and bounded.
OR
Let Y and Z be subspaces of a normed space X and suppose that Y is closed and is a proper subset of Z. Then for every real number θ in the interval (0, 1) there is a z ∈ Z such that ||z|| = 1, ||z−y|| ≥ θ for all y ∈ Y.
9.
If a normed space x is finite dimensional, then prove that every linear operator on x is bounded.
OR
Let T : D(T) → y be a bounded linear operator, where D(T) lies is a normed space x and y is a Banach space. Then prove that T has an extension T̄ : D(T) → Y where T̄ is a bounded linear operator of norm ||T̄|| ||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 lr.
11.
Let x be an inner product space and M = ϕ a convex subset which is complete. Then prove that for every given x ∈ X there exists a unique y ∈ M such that δ = infy∈M ||x − y|| = ||x − y||
OR
State and prove Riesz's theorem for functionals on Hilbert spaces.