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Politecnico di Milano , Mathematical Engineering Real and Functional Analysis – Theory Prof. F. Punzo, G. Verzini, January 20, 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) be a measurable space and assume that, for every n ∈ N, the function un :X → R is measurable. Prove that the function sup nun is Lebesgue measurable too. 1 Question 2. [4 points] Considerfn,n ∈ N andf, measurable functions from a measure space (X, M,µ ) to R. Write the following definitions: 1. {fn}n converges to f in measure; 2. {fn}n converges to f in the L1-sense. What is the relation between 1. and 2.? No proofs are required. 2 Question 3. [4 points] Let X be a Banach space, and let X ∗∗ denote its bidual. 1. Provide the definition of the canonical (evaluation) map τ :X →X ∗∗. 2. List the principal properties of τ. 3. What does it mean that X is reflexive? No proofs are required. 3 Question 4. [4 points] State and prove the Riesz representation theorem in Hilbert spaces. 4