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- University
- Politecnico di Milano
- Degree programme
- Mathematical Engineering
- Subject
- Real and functional analysis
- Academic year
- 2023-2024
- Classification
- Exam · Full exam
- Content
- Exam paper only
- Original format
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20 06 2024 E T
Full exam for Real and functional analysis in the Mathematical Engineering degree programme at Politecnico di Milano. The document covers: Politecnico di Milano , Mathematical Engineering Real and Functional Analysis – Theory Prof. F. Punzo, G. Verzini, June 20, 2024 Q1 Q2 Q3 Q4 Surname/Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Id. No. . . .
Real and functional analysisFull exam
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Full exam for Real and functional analysis in the Mathematical Engineering degree programme at Politecnico di Milano. The document covers: Politecnico di Milano , Mathematical Engineering Real and Functional Analysis – Theory Prof. F. Punzo, G. Verzini, June 20, 2024 Q1 Q2 Q3 Q4 Surname/Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Id. No. . . .
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Politecnico di Milano , Mathematical Engineering Real and Functional Analysis – Theory Prof. F. Punzo, G. Verzini, June 20, 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] Given a metric space ( X, d), write the definition of its Borel σ−algebra B(X). Provide classes of Borel sets. Characterize B(R), B(R), B(RN). No proofs are required. 1 Question 2. [4 points] State and prove the Lebesgue (dominated convergence) theorem. 2 Question 3. [4 points] State and prove the H¨ older inequality. 3 Question 4. [4 points] Write the statement of the Banach-Alaoglu theorem (both versions). Provide an example of its application. No proofs are required. 4
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