26 01 2022 E TS

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Real and Functional Analysis Master Degree Program in Mathematical Engineering, a.y. 2021/22 26 January 2022 Answers and solutions can be written in English or in Italian. Theory Question 1. (4 points) Let (X,A,µ ) be a measure space and denote byM(X,A) the vector space of measurable functions. 1. Let{fn}⊂M (X,A). Show that lim inf n→+∞ fn∈M (X,A). 2. State and prove Fatou’s Lemma. Question 2. (4 points) 1. Let f : [a,b ]→ R be a function. Write the definitions of: variation off relative to a certain partition of [a,b ]; total variation of f on the interval [a,b ]; function of bounded variation. Exhibit an example of functionf∈BV ([a,b ]) and of functiong̸∈BV ([a,b ]). 2. Is it true thatf∈BV ([a,b ]) is Riemann integrable? Justify the answer. Question 3. (4 points) Let X,Y be two Banach spaces. 1. Write the definition of (linear) compact operatorK :X→Y . 2. Let K :X→Y be a linear and continuous operator. Recall thatK is said to be weak-strong continuous whenever xn⇀x ⇒K(xn)→K(x), as n→ +∞. Which is the relation between compactness ofK and weak-strong continuity ofK? If necessary, you can specify further hypotheses forX. Question 4. (4 points) Let (H,⟨·,·⟩) be a separable Hilbert space. 1. Suppose thatλ∈ R is an eigenvalue of a linear compact operatorK :H→H. What can we conclude about the dimension of the eigenspaceVλ? Justify the answer. 2. State and prove the theorem concerning minimal distance from convex closed subsets in Hilbert spaces. 1 Exercises Exercise 1. (7 points) Consider the Lebesgue measure space(R,L(R), λ). Let fn : R→ R be the sequence of functions defined by fn(x) = 5− 1 nx2nχ[−1,1](x), x ∈ R, for anyn∈ N. 1. Study the convergence almost everywhere of{fn}n∈N. Does the sequence converge pointwisely everywhere? Justify the answer. 2. Study the convergence…