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Politecnico di Milano , Mathematical Engineering Real and Functional Analysis – Theory Profs. G. Verzini (A-L), F. Punzo (M-Z). February 14, 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] (i) For (X, M), (Y, N ) measurable spaces, write the definition of ( M, N )–measurable function. (ii) Let S ⊂ N be such that N is generated by S. State and prove an equivalent condition for the ( M, N )– measurability of a function, in terms of S. 1 Question 2. [4 points] (i) State the characterization of absolutely continuous functions in [ a, b]. No proofs are required. (ii) Exhibit a function f which is continuous in some interval [ a, b], but not absolutely continuous in [ a, b]. (iii) Let {fn} ⊂ AC([a, b]), fn → f as n → +∞ uniformly in [ a, b]. Prove or disprove by means of a counterexample the following: f ∈ AC([a, b]). 2 Question 3. [4 points] State and prove the Banach-Steinhaus theorem (or uniform boundedness principle). 3 Question 4. [4 points] Let H be a Hilbert space and f ∈ H. Under suitable assumptions on T : H → H, that have to be specified, discuss the solvability of the equation (for the unknown x ∈ H) x − T x = f. No proofs are required. 4