25 01 2023 E TS

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Mathematical Engineering - A.Y. 2022-23 Real and Functional Analysis - Exam with Solutions- January 25, 2023 Answers and solutions can be written in English or in Italian. Theory Question 1. (4 points)(i) State and prove the property of continuity of measure along mono- tone decreasing sequences{En} of measurable subsets. (ii) Does the property hold, ifE1 (or some of the setsEn) has infinite measure? If not, provide a counterexample. Solution. See Lecture 2. Question 2. (4 points)State and prove the Fatou’s Lemma. Solution. See Lecture 7. Question 3 (4 points) (i) Write the definition of open mapping. State the Open Mapping theorem. (ii) State and prove the Inverse Bounded Mapping Theorem. Solution. See Lecture 18. Question 4 (4 points)LetX be a Banach space. (I) Write the definitions of weak convergence and of (strong) convergence for a sequence{xn}⊂ X. (II) Consider now the following properties: (a) xn ⇀x (weakly) inX; (a’) {xn} possesses a weakly convergent subsequence; (b){xn} is bounded; (c) xn→x (strongly) inX. (i)Does (a)imply (b)? (ii)Does (b)imply (a′)(ifnecessary, underadditionalassumptions)? (iii) Does (a) imply (c)? (iv) Does (c) imply (a)? For questions 4.(II)(i)-(iv), justify the answers, only quoting some theorems or briefly dis- cussing a counterexample. No proofs are required. Solution. See Lectures 14, 21, 22. 2 Exercises Exercise 1. Consider the measure space ([0, +∞),L([0, +∞))) with the Lebesgue measure. Define the sequence of functions{fn}n∈N by fn(x) = sin2(x) 1 +x χ[0,n](x), x ∈ R, n ∈ N. (1) Prove thatfn∈Lp([0, +∞)) for anyn∈ N and anyp∈ (1, +∞). (2) Study the convergence a.e. of the sequence{fn}n∈N. (3) Study the convergence inLp([0, +∞)) of the sequence{fn}n∈N for p∈ (1, +∞). Solution. (1) Letn∈ N and p∈ (1, +∞), we have ∫ +∞ 0 |fn(x)|p dx = ∫ n 0…