11 07 2024 E TS

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Politecnico di Milano , Mathematical Engineering Real and Functional Analysis – Exercises Prof. F. Punzo, G. Verzini, July 11, 2024 E1 E2 E3 Surname/Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Id. No. . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Solutions must be written ONL Y on these sheets, under the exercise and in the back.] [Solutions can be written in English or in Italian.] Exercise 1. [6 points] 1. Consider the sequence gn(x) = 2−nxn, x ∈ R. (a) Compute the set D of pointwise convergence of the sequence {gn}n∈N. (b) Then compute the pointwise limit g : D → R of the sequence {gn}n∈N. (c) Does {gn}n∈N converge in L∞(D) to g? 2. Consider the sequence fn(x) = n2 log(1 + 2 nx) x + n , x ∈ (1, +∞). (a) Compute the pointwise a.e. limit f of the sequence {fn}n∈N. Does f belong to L1(1, +∞)? (b) Compute lim n→∞ R ∞ 1 fn(x) dx. Does the sequence {fn}n∈N converge in L1(1, +∞)? Solution. (1a) Observing that gn(x) =  x 2 n , we get that gn(x) converges if and only if x ∈ (−2, 2]. (1b) Let x ∈ (−2, 2], then we have gn(x) → g(x) = ( 1, if x = 2, 0, if − 2 < x < 2. (1c) Since uniform a.e. convergence implies pointwise a.e. convergence and {gn}n∈N converges to g pointwisely, then g is the unique candidate limit for L∞-convergence. Observe that |gn(x) − g(x)| = ( 0, if x = 2, 2−nxn, if − 2 < x < 2. Then, we get ∥gn − g∥∞ = esssupx∈(−2,2) |gn(x) − g(x)| = 1 ̸→ 0 as n → +∞. Therefore {gn}n∈N does not converge to g in L∞(D). (2a) Let x ∈ (1, +∞). We get fn(x) = n2 log(1 + 2 nx) x + n = n2 log(1 + 2 nx) n(1 + x n) → 2 x =: f(x), as n → +∞. We notice that f(x) = 2 x is not an L1(1, +∞) function. (2b) Since fn is a nonnegative function for every n ∈ N, we can apply Fatou’s Lemma and get lim inf…