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.

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## Functional Analysis MCQs Quiz.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.

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