06 09 2022 E TS

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Real and Functional Analysis Master Degree Program in Mathematical Engineering, a.y. 2021/22 6/9/2022 Answers and solutions can be written either in English or in Italian. Theory Question 1. (4 points)Let (X,A,µ ) be a (complete) measure space. 1. Define the convergence in mean and the convergence in measure. 2. Prove that the convergence in mean implies the convergence in measure. Is the converse true? Justify the answer. Question 2. (4 points)Let f∈L1([a,b ]). 1. Give the definition of Lebesgue point forf and show an example of non-Lebesgue point. 2. State and prove the first fundamental Theorem of Calculus. Question 3. (4 points)Let (X,d ) be a metric space. 1. Define completeness and separability forX. 2. Give an example of separable infinite-dimensional normed vector space. Prove thatL∞([a,b ]) is not separable. Question 4. (4 points)Let (X,∥·∥ ) be a normed vector space. 1. State the Hahn-Banach theorem. 2. Define the canonical mapτ betweenX and its dualX∗, prove thatτ is an isometry, and give the definition of reflexivity. 1 Exercises Exercise 1. (7 points)Given the sequence of functions fn(x) = n sin(x) 1 +n2√x, x ∈ [0, 1], n ∈ N, 1. study the convergence a.e. of the sequence{fn}n∈N in [0, 1] and prove that, for anyn∈ N, the function fn is integrable in[0, 1]; 2. study the convergences inL1([0, 1]) and in measure of the sequence{fn}n∈N in [0, 1]; 3. compute, if it exists finite, the following limit lim n→+∞ ∫ 1 0 fn(x) dx. Solution. (1) To study the pointwise (a.e.) convergence of{fn}n, we distinguish two cases. (i) x = 0. In this case, sincefn(0) = 0 for everyn∈ N, we immediately get lim n→+∞ fn(0) = 0. (ii) 0<x ≤ 1. In this case, sincesin(x),√x> 0, we have fn(x) = n sin(x) 1 +n2√x∼ n sin(x) n2√x = 1 n· sin(x)√x as n→ +∞; from this, we infer thatfn(x)→ 0 as n→ +∞.…