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Functional Analysis (325b/525b) Practice Final exam questions. Michael Magee, michael.magee@yale.edu. Question 1. a. Prove a normed space is complete if and 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). 1) Prove that. −→. 0 is unique in any normed space. Solution of 1) Let us suppose that there are 2 zeros 01 and 02. Then. 01 = 01 + 02 = 02 + 01(a) State Bessel's inequality. Solution. If H is a Hilbert space and (φn)n≥1 is an orthonormal sequence, then. ∞. Functional analysis is the analysis of infinite dimensional vector spaces, typically spaces of Both questions are answered in the following sections. ) is a Banach space. (5 points). Solution: We show that (X,. ∞. ) is a Banach space Functional Analysis Exam Solutions. Question 1. (i) A normed vector space V is a Banach space if it is complete, that is to. 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.