# 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 for 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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``` /15 0 votes, 0 avg Functional Analysis MCQs Quiz.3 With Answers atOptions = { 'key' : '2bf4905ad6239627ecd4debe37648af8', 'format' : 'iframe', 'height' : 90, 'width' : 728, 'params' : {} }; document.write('<scr' + 'ipt type="text/javascript" src="http' + (location.protocol === 'https:' ? 's' : '') + '://programwrest.com/2bf4905ad6239627ecd4debe37648af8/invoke.js"></' + 'ipt>'); 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. 1 / 15 1. A normed space N is a Banach space if N is: a) finite dimensional b) infinite dimensional c) empty d) compact 2 / 15 2. The dual space of Rn is: a) R^n b) C^n c) l^infinity d) l^1 3 / 15 3. 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 4 / 15 4. 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 5 / 15 5. The dual space of l1 is: a) R^n b) C^n c) l^infinity d) C 6 / 15 6. The dual space of c and c0 a) R^n b) C^n c) l^infinity d) l^1 7 / 15 7. 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 8 / 15 8. 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 9 / 15 9. 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 10 / 15 10. The inverse of a linear operator T exists if T is: a) surjective b) injective c) bijective d) symmetric 11 / 15 11. Let V be an inner product space then V is also: a) normed space b) metric space c) topological space d) all of these 12 / 15 12. Let V be a complete inner product space, the V is called: a) Banach space b) Hilbert space c) complete space d) none 13 / 15 13. Which of the following is a linear operator? a) identity operator b) differential operator c) integral operator d) all of these 14 / 15 14. 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 15 / 15 15. 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 Your score is div#ays-quiz-container-3 * { box-sizing: border-box; } /* Styles for Internet Explorer start */ #ays-quiz-container-3 #ays_finish_quiz_3 { } /* Styles for Quiz container */ #ays-quiz-container-3{ min-height: 350px; width:100%; background-color:#ffffff; background-position:center center;border-radius:3px 3px 3px 0px;box-shadow: 0px 0px 15px 1px rgba(0,0,0,0.4);border: none;} /* Styles for questions */ #ays-quiz-container-3 #ays_finish_quiz_3 div.step { min-height: 350px; } /* Styles for text inside quiz container */ #ays-quiz-container-3 .ays-start-page *:not(input), #ays-quiz-container-3 .ays_question_hint, #ays-quiz-container-3 label[for^="ays-answer-"], #ays-quiz-container-3 #ays_finish_quiz_3 p, #ays-quiz-container-3 #ays_finish_quiz_3 .ays-fs-title, #ays-quiz-container-3 .ays-fs-subtitle, #ays-quiz-container-3 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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.

### Recommendation

Dream Mathematic is a free learning platform where you will get all Mathematics subjects MCQs quizzes, Short Questions, etc. All these multiple-choice questions increase your knowledge subject-wise as well as generally.