Functional Analysis MCQs Quiz.3 and Short Questions


There, we provide you with a Functional Analysis MCQs Quiz, and Short Questions with answers for better test preparations. All these Functional Analysis Quizzes are also available in pdf and you will be able to download them free. Most questions are explained well for your exam practice. In this quiz we are providing you with multiple choice questions on different topics, linear operators bounded or unbounded linear operators, bench space, normal space, etc.  

Here, mostly past papers MCQ b/c most of the university entrance tests every year include past papers exam questions. Solve these multiple choice questions carefully because it’s most crucial for your university entrance tests.

After Solving this multiple choice questions quiz you will get practice questions that are easier for you. We ensure that after solving this MCQs quiz you solve these short questions easily by using the above concepts. 

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

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2. The inverse of a linear operator T exists if T is:

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3. Any two norms on a linear space N are equivalent if N is:

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4. Let the dimension of a normed space N is n, then the dimension of its dual space N* is:

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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:

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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

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7. Which of the following is a normed space which is not inner product space?

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8. Let N is a normed space in which every closed and bounded subset is compact then N is:

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9. A normed space N is a Banach space if N is:

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10. The dual space of Rn is:

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11. The dual space of l1 is:

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12. Let V be an inner product space then V is also:

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13. Let V be a complete inner product space, the V is called:

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14. Let N be a finite-dimensional normed space, then N is isomorphic to

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15. Which of the following is a linear operator?

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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.