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Politecnico di Milano , Mathematical Engineering Real and Functional Analysis – Theory Prof. F. Punzo, G. Verzini, February 13, 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] State and prove the theorem about regularity of the Lebesgue measure. 1 Question 2. [4 points] Write the following definitions: a) absolutely continuous function in [ a, b]; b) W 1,1([a, b]) and weak derivative of a function defined in [ a, b]; What is the relation between W 1,1([a, b]) and the set of absolutely continuous functions in [ a, b]? No proofs are required. 2 Question 3. [4 points] Let X be a Banach space and let {xn}n∈N⊂ X, x∈ X. a) write the following definition: xn ⇀ x as n→ +∞ weakly in X; b) assuming that xn ⇀ x as n→ +∞ weakly in X, state and prove what can be deduced about ∥x∥. 3 Question 4. [4 points] Let H be a Hilbert space, T∈L (H). Write the following definitions: a) T is compact; b) resolvent set of T ; c) spectrum of T . State the spectral theorem. No proofs are required. 4