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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 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). Functional analysis is the analysis of infinite dimensional vector spaces, typically spaces of func- tions of all possible kinds, and of linear operators on 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 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. Problems – Chapter 1. Problem 5.1. Solutions to problems. Solution 5.1 (5.1). (5) Show that the integral is a continuous linear functional. 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.