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Politecnico di Milano , Mathematical Engineering Real and Functional Analysis – Theory Prof. F. Punzo, G. Verzini, January 24, 2024 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 λ∗ be the outer measure on R. Starting from it, describe briefly the construction of the Lebesgue σ−algebra L(R) and of the Lebesgue measure λ on R. No proofs are required. 1 Question 2. [4 points] State and prove the monotone convergence theorem. 2 Question 3. [4 points] Let X be a Banach space, and let X ∗ denote its dual. Let {Ln}n ⊂ X ∗, L ∈ X ∗. Write the following definitions: 1. {Ln}n converges to L in the weak sense as n → +∞; 2. {Ln}n converges to L in the weak ∗ sense as n → +∞. Which is the relation between 1. and 2.? No proofs are required. 3 Question 4. [4 points] State and prove the projection theorem in Hilbert spaces. 4