Esame completo di Real and functional analysis per il corso di Mathematical Engineering presso Politecnico di Milano. Materiale proveniente dall’archivio storico Studwiz e classificato per la consultazione online.
Esame completo di Real and functional analysis per il corso di Mathematical Engineering presso Politecnico di Milano. Materiale proveniente dall’archivio storico Studwiz e classificato per la consultazione online.
Qualità dell’importazione: il testo è stato estratto direttamente dal documento originale.
Passaggi rappresentativi riconosciuti nelle diverse parti del materiale. Il testo completo resta presente nella pagina per la ricerca, mentre l’anteprima compatta rende più semplice la lettura.
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
Prima pagina del documento.
