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Politecnico di Milano , Mathematical Engineering Real and Functional Analysis – Theory Profs. F. Punzo, G. Verzini. September 9, 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 ( X, M, µ) denote a measure space. Prove the continuity of the measure along monotone sequences, either increasing or decreasing, under suitable further assumptions. 1 Question 2. [4 points] Consider fn, n ∈ N and f, measurable functions from a measure set ( 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. Which is the relation between 1. and 2.? No proofs are required. 2 Question 3. [4 points] State and prove the Banach-Steinhaus theorem (or uniform boundedness principle). 3 Question 4. [4 points] Consider a linear operator K : X → Y , with X and Y Banach spaces. Write the following definitions: 1. K is compact; 2. K is finite rank. Which is the relation between 1. and 2.? No proofs are required. 4