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Functional analysis is the analysis of infinite dimensional vector spaces, typically spaces of func- tions of all possible kinds, and of linear operators on The classic solution of this problem for a space X which is compact Hausdorff (or even normal) is known as the Riesz representation theorem, and it depends H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential solution of the equation tan t = −1/t (which has an infinite sequence ofSolution of 2) Let {xn} be a Cauchy sequence in C(X); that is for any ε > 0 there exists N0 ∈ N such that. (1) n, m ≥ N0. FUNCTIONAL ANALYSIS: NOTES AND PROBLEMS. Abstract. These are the notes prepared for the course MTH 405 to be offered to graduate students at IIT Kanpur. Problem 1. Prove that any ball in a normed space X is convex. Solution. Let B(x0;r) be any ball of radius r > 0 centered at x0 ∈ X, and x, y ∈ B(x0; r). Problems – Chapter 1. Problem 5.1. Solutions to problems. Solution 5.1 (5.1). (5) Show that the integral is a continuous linear functional. Then, (U, ⋅ ) is a Banach space as well. 1.2. Examples of Banach spaces. Example 1. Let B(T) be a space consisting of all bounded maps