20 01 2025 E TS

Pagina 1

Politecnico di Milano , Mathematical Engineering Real and Functional Analysis – Exercises Prof. F. Punzo, G. Verzini, January 20, 2025 E1 E2 E3 Surname/Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Id. No. . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Solutions must be written ONL Y on these sheets, under the exercise and on the back.] [Solutions can be written in English or in Italian.] Exercise 1. [5 points] Consider the sequence fn(x) = n2 + 1 n2(x2 − 2) + (n2 + 1)x , x ∈ (2, +∞). 1. Compute the pointwise limit f of the sequence {fn}+∞ n=1; 2. Does the sequence {fn}+∞ n=1 converge in L1(2, +∞)? 3. Compute lim n→+∞ R +∞ 2 fn(x) dx. Solution. (1) Let x ∈ (2, +∞). We get fn(x) = n2 + 1 n2(x2 − 2) + (n2 + 1)x = n2 + 1 (n2 + 1)( n2 n2+1(x2 − 2) + x) → 1 x2 + x − 2 =: f(x), as n → +∞. (2) Since L1 convergence implies the existence of a subsequence which converges almost everywhere and the sequence {fn}n∈N converges to f pointwisely, then f is the unique candidate limit for L1-convergence. We proceed by Dominated Convergence Theorem applied to the sequence of functions {fn}n∈N. It is enough to show that there exists a function g ∈ L1(2, +∞) such that |fn| ≤ g for every n ∈ N. We observe that |fn(x)| = n2 + 1 n2(x2 − 2) + (n2 + 1)x = n2 + 1 (n2 + 1)( n2 n2+1(x2 − 2) + x) ≤ 1 1 2(x2 − 2) + x = 2 x2 + 2x − 2 =: g(x) for each n ∈ N and x ∈ (2, +∞). Therefore, the sequence {fn}n∈N converges to f in L1(2, +∞). (3) By the previous item, we have lim n→+∞ Z +∞ 2 fn(x)dx = Z +∞ 2 f(x)dx = Z +∞ 2 1 x2 + x − 2dx = lim x→+∞ 1 3[log(x − 1) − log(x + 2) + log(4)] = 1 3 log(4). 1 Exercise 2. [5 points] Consider the sequence x(k) n = ( 1 3√ k , for n ≤ k ≤ 2n, 0, otherwise. Study 1. the pointwise convergence…