Functional Analysis-Ii

Course: M.A.M.Sc. Code: GH-499 Branch: Maths Year: 2022

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Roll No. ........................................
Total No. of Questions : 11
[Total No. of Printed Pages : 10

GH-499

M.A./M.Sc. IVth Semester (Reg./Pvt./ATKT)
Examination, 2022

Maths

Paper - Compulsory

Functional Analysis-II

Time: 3 Hours]
[Maximum Marks: Reg=85 Pvt.=100

Note- Attempt all the questions from Section "A", "B" and "C". Internal choice is given where necessary. Marks are indicated. Mathematical notations are in their usual meanings.

SECTION - 'A'

Objective Type Questions

(vii)
A real valued functional p on a vector space X is subadditive if for all x, y ∈ X:
  1. p(x+y)≤p(x)+p(y)
  2. p(x+y)>p(x)+p(y)
  3. p(x+y)≤p(x)+p(y)
  4. p(x+y)≥p(x)+p(y)
(viii)
If T is an operator on a Hilbert space H, such that ||T*x||=||Tx|| ∀x, then T is:
  1. Normal
  2. Unitary
  3. Positive
  4. None of these
(ix)
Which of the following is true
  1. Every normed space is reflexive
  2. Every normed space is complete
  3. Every normed space which is reflexive is complete
  4. Every reflexive space is complete
(x)
In a metric space X, if for any subset M, its closure M̄ has no interior point, then M is said to be:
  1. Rare or nowhere dense in
  2. Of the category first
  3. Of the category second
  4. None of these
(xi)
In a metric space X, if for any subset M, M is the union of countably many sets, each of which is rare in X, then M is said to be:
  1. Complete in X
  2. Of first category in X
  3. Of second category in X
  4. None of these
(xii)
Let X and Y be normed spaces, a sequence of operators (Tn), Tn ∈ B(X, Y) is said to be uniformly operator convergent, if ∃ an operator T, such that.
  1. ||Tn - T|| → 0 for all x ∈ X
  2. ||Tn - T|| → 0
  3. If ||(Tn)(x) - f(Tn)(x)|| → 0 for all x ∈ X and f ∈ y'
  4. None of these
(xiii)
Let X and Y be normed spaces, a sequence of operators (Tn), Tn ∈ B(X,Y) is said to be strongly operator convergent, if ∃ an operator T, such that:
  1. ||Tnx - Tx|| → 0 for all x ∈ X
  2. ||Tn - T|| → 0
  3. If ||(Tn)(x) - f(Tn)(x)|| → 0 for all x ∈ X for all f ∈ y'
  4. None of these
(xiv)
The adjoint of a bounded linear operator in Hilbert spaces is:
  1. Unique
  2. Not unique
  3. Cant say
  4. None of these
(xv)
A bounded linear operator T: H → H where H is a Hilbert space then T is unitary, if.
  1. T*=T
  2. TT*=T
  3. T*=T-1
  4. None of these

SECTION - 'B'

Short Answer Type Questions

2.
Define Hilbert adjoint operator and self adjoint operator.
OR
3.
Define unitary and normal operators.
OR
Define partial ordered set, chain and only state Zorn's lemma.
In a normed space X, if ∀x ∈ X
||x|| = Sup |f(x)|
and if xo is
f≠0
Diagram for Question
and if xo is such that f(xo) = 0 for all x ∈ X' then show that
4.
Define adjoint operator on normed spaces and show that
(T1 + T2)* = T1* + T2*
where T1, T2 ∈ B(X,Y) are adjoint operators.
5.
Show that every Hilbert space is reflexive.
Define the terms rare, meager and nonmeager w.r.to categories.
OR
6.
In short discuss strong and weak convergence in normed spaces.
OR
In short discuss strong and weak convergence of the sequence of functionals.

SECTION - 'C'

Long Answer Type Questions

7.
Show that the Hilbert adjoint operator T* of T with the usual meaning T: H1 → H2, defined by <Tx, y> = <x, T*y> where x ∈ and y ∈ H2, exists and unique and also a bounded linear operator with norm.
H1 → H2, α is any scalar then show that:
  1. αT* = αT*
  2. T** = T
  3. (ST)* = T*S*
  4. ||TT*||=||T*T||=||T||2
  5. T*T = 0 ⇔ T = 0
8.
State and prove Hahn-Banach theorem for real linear spaces.
OR
State and prove Hahn - Banach theorem for normed linear spaces.
9.
Prove that the adjoint operator T*: Y' → X' (where X and Y are normed space) is linear, bounded and ||T*|| = ||T||.
OR
Show that
  1. a normed space X which is reflexive is complete.
  1. Every finite dimensional normed space is reflexive.
10.
State and prove Baire's category theorem for complete metric space.
If S and T are bounded linear operators on Hilbert spaces X → X and if only if
  1. The sequence {||xn||} is bounded
  2. ∀ element f of a total subset M⊆X' we have f(xn) → f(x)
11.
State and prove closed graph theorem.
OR
State and prove open mapping theorem.
GH-499
(1)
P.T.O.