Functional Analysis MCQs Quiz.3 and Short Questions

Functional Analysis Important MCQs / By lernerusman

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Functional Analysis MCQs Quiz.3;

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                                                                                                                                                            Functional Analysis MCQs Quiz.3 With Answers
                    In the Functional Analysis MCQs Quiz, we are sharing Questions with you on Linear Operator, Inner product space, Banach space, and Linear Functional Analysis MCQs questions to help us to increase our knowledge.
                                                                                                                                            
                
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                                                                                                    1. The dual space of c and c₀
                            
                                                                                                                a) R^n                            
                                                            b) C^n                            
                                                            c) l^infinity                            
                                                            d) l^1                            
                                                                                                                                                                                                                                                                    
                                                                                                    
                                                                            
                                                                            
                                            
                
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                                                                                                    2. The inverse of a linear operator T exists if T is:
                            
                                                                                                                a) surjective                            
                                                            b) injective                            
                                                            c) bijective                            
                                                            d) symmetric                            
                                                                                                                                                                                                                                                                    
                                                                                                    
                                                                            
                                                                            
                                            
                
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                                                                                                    3. Any two norms on a linear space N are equivalent if N is:
                            
                                                                                                                a) finite dimensional                            
                                                            b) infinite dimensional                            
                                                            c) complete space                            
                                                            d) Banach space                            
                                                                                                                                                                                                                                                                    
                                                                                                    
                                                                            
                                                                            
                                            
                
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                                                                                                    4. Let the dimension of a normed space N is n, then the dimension of its dual space N* is:
                            
                                                                                                                a) less than n                            
                                                            b) equal to n                            
                                                            c) greater then n                            
                                                            d) none                            
                                                                                                                                                                                                                                                                    
                                                                                                    
                                                                            
                                                                            
                                            
                
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                                                                                                    5. Let N and M be Banach spaces and T : N → M be a bijective continuous linear operator.
Then T is a:
                            
                                                                                                                a) open mapping                            
                                                            b) closed mapping                            
                                                            c) homomorphism                            
                                                            d) compact mapping                            
                                                                                                                                                                                                                                                                    
                                                                                                    
                                                                            
                                                                            
                                            
                
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                                                                                                    6. Let V be a normed space and U be a subspace of V. Let f₀ be a bounded linear functional on V with norm || f₀ ||. Then f₀ has a continuous linear extension f defined on V such that || f || = || f₀ || is a statement of
                            
                                                                                                                a) Reiez’s Lemma                            
                                                            b) Cauchy’s Theorem                            
                                                            c) Hahn-Banach Theorem                            
                                                            d) Taylor’s Theorem                            
                                                                                                                                                                                                                                                                    
                                                                                                    
                                                                            
                                                                            
                                            
                
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                                                                                                    7. Which of the following is a normed space which is not inner product space?
                            
                                                                                                                a) R^n                            
                                                            b) C^n                            
                                                            c) l_3                            
                                                            d) l_4                            
                                                                                                                                                                                                                                                                    
                                                                                                    
                                                                            
                                                                            
                                            
                
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                                                                                                    8. Let N is a normed space in which every closed and bounded subset is compact then N is:
                            
                                                                                                                a) finite dimensional                            
                                                            b) infinite dimensional                            
                                                            c) Banach space                            
                                                            d) compact space                            
                                                                                                                                                                                                                                                                    
                                                                                                    
                                                                            
                                                                            
                                            
                
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                                                                                                    9. A normed space N is a Banach space if N is:
                            
                                                                                                                a) finite dimensional                            
                                                            b) infinite dimensional                            
                                                            c) empty                            
                                                            d) compact                            
                                                                                                                                                                                                                                                                    
                                                                                                    
                                                                            
                                                                            
                                            
                
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                                                                                                    10. The dual space of Rⁿis:
                            
                                                                                                                a) R^n                            
                                                            b) C^n                            
                                                            c) l^infinity                            
                                                            d) l^1                            
                                                                                                                                                                                                                                                                    
                                                                                                    
                                                                            
                                                                            
                                            
                
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                                                                                                    11. The dual space of l¹ is:
                            
                                                                                                                a) R^n                            
                                                            b) C^n                            
                                                            c) l^infinity                            
                                                            d) C                            
                                                                                                                                                                                                                                                                    
                                                                                                    
                                                                            
                                                                            
                                            
                
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                                                                                                    12. Let V be an inner product space then V is also:
                            
                                                                                                                a) normed space                            
                                                            b) metric space                            
                                                            c) topological space                            
                                                            d) all of these                            
                                                                                                                                                                                                                                                                    
                                                                                                    
                                                                            
                                                                            
                                            
                
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                                                                                                    13. Let V be a complete inner product space, the V is called:
                            
                                                                                                                a) Banach space                            
                                                            b) Hilbert space                            
                                                            c) complete space                            
                                                            d) none                            
                                                                                                                                                                                                                                                                    
                                                                                                    
                                                                            
                                                                            
                                            
                
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                                                                                                    14. Let N be a finite-dimensional normed space, then N is isomorphic to
                            
                                                                                                                a) N* (first dual space)                            
                                                            b) N** (second dual space)                            
                                                            c) N*** (third dual space)                            
                                                            d) all of these                            
                                                                                                                                                                                                                                                                    
                                                                                                    
                                                                            
                                                                            
                                            
                
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                                                                                                    15. Which of the following is a linear operator?
                            
                                                                                                                a) identity operator                            
                                                            b) differential operator                            
                                                            c) integral operator                            
                                                            d) all of these                            
                                                                                                                                                                                                                                        
                                                                                                    
                                                                            
                                                                            
                                            
                
            
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Functional Analysis Short Questions;

Question No.1

(a) Prove a normed space is complete if and only if every absolutely convergent series is convergent

(b) Prove ` p is complete for 1 ≤ p < ∞. (You can assume it is a well-defined normed vector space.)

Question No.2

(a) State and prove the Hahn-Banach Theorem. (You can and should use Zorn’s lemma.)

(b) Show that if X and Y are normed vector spaces and T ∈ B(X, Y ) then ||T|| = ||T ∗||.

Question No.3

(a) Show that a normed vector space is finite-dimensional if and only if its closed unit ball is compact.

Question No.4

(a) State and prove the Baire Category Theorem in some form.

(b) Let C R([0, 1]) denote the real-valued continuous functions on [0, 1]. By considering for each n ≥ 1 the set Fn = {f ∈ C([0, 1]) : |f(x) − f(y)| /|x − y| ≤ n ∀ 0 ≤ x, y ≤ 1, x not = y }, deduce that the continuous nowhere-differentiable functions are dense in C^R([0, 1]).

Question No.5

a) State and prove the open mapping Theorem, and deduce the closed graph Theorem. (You may assume any version of the Baire Category Theorem provided it is stated clearly.)

(b) State and prove the Arzel`a-Ascoli Theorem.

Question no.6

(a) Sketch a proof of Montel’s Theorem: a uniformly bounded set of holomorphic functions on an open set Ω ⊂ C contains a subsequence that is uniformly convergent on all compact subsets of Ω.

(b) State and prove the Alexander sub-base Theorem.

Question No.7

(a) Using the sub-base Theorem, show that [0, 1] is compact. (Do not use Heine-Borel or Bolzano-Weierstrass).

(b) State the ‘Geometric’ Hahn Banach Theorem. Prove that a convex subset of a normed space is closed if and only if it is weakly closed.