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Politecnico di Milano , Mathematical Engineering Real and Functional Analysis – Theory Profs. F. Punzo, G. Verzini, June 19, 2025 Q1 Q2 Q3 Q4 Surname/Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Id. No. . . . . . . . . . . . . . . . . . . . . . . . . . . . . [Answers must be written ONL Y on these sheets, under the question and in the back.] [Answers can be written in English or in Italian.] Question 1. [4 points] Let (X, M), ( Y, N ) be measurable spaces, S ⊂ N be a family generating N , and let f : X → Y be a function. State and prove the theorem that relates the measurability of f to the measurability of the sets f −1(E), for every E ∈ S . 1 Question 2. [4 points] Let f : [a, b] → R. (i) Write the definition of Lebesgue point. (ii) What can be said about the Lebesgue measure of the set A = {x ∈ [a, b] : x is not a Lebesgue point of f }? (iii) State the first fundamental theorem of calculus for functions in L1((a, b)). No proofs are required. 2 Question 3. [4 points] Let X, Y be Banach spaces with Y infinite-dimensional. State and prove the theorem relating compact (linear) operators and finite rank operators between X and Y . 3 Question 4. [4 points] Let X be a Banach space and let {xn} be a sequence that converges weakly to x in X. (i) Is it true that the sequence {xn}n is necessarily bounded in X? (ii) What can be said about lim inf ∥xn∥? (iii) If {Ln}n ⊂ X ∗ converges strongly to L, what can be said about the sequence {Ln(xn)}n? (iv) If T ∈ L(X), what can be said about the sequence {T (xn)}n? No proofs are required. 4