18.102: PROBLEMS FOR TEST1 – 11 OCTOBER, 2007FIRST CORRECTED VERSION, THANKS TO URS NIESENRICHARD MELROSEThese 10 questions will reappear on the test on Thursday october 11 when youwill be asked to prove 3 of them (my choice, not yours), without using notes or thebook. However you may use the earlier problems to prove the later ones – even inthe test itself. I will not change this except to make corrections if any are needed.Recall that the objective is to give an answer which is clear, concise and complete!(1) Recall Lebesgue’s Dominated Convergence Theorem and use it to show thatif u ∈ L2(R) and v ∈ L1(R) then(1)limN→∞Z|x|>N|u|2= 0, limN→∞Z|CNu − u|2= 0,limN→∞Z|x|>N|v| = 0 and limN→∞Z|CNv − v| = 0.where(2) CNf(x) =N if f(x) > N−N if f(x) < −Nf(x) otherwise.(2) Show that step functions are dense in L1(R) and in L2(R) (Hint:- Lookat Q1 and think about f − fN, fN= CNfχ[−N,N]and its square. So itsuffices to show that fNis the limit in L2of a sequence of step functions.Show that if gnis a sequence of step functions converging to fNin L1thenCNχ[−N,N]is converges to fNin L2.) and that if f ∈ L1(R) then there is asequence of step functions unand an element g ∈ L1(R) such that un→ fa.e. and |un| ≤ g.(3) Show that L1(R) and L2(R) are separable, meaning that each has a count-able dense subset.(4) Show that the minimum and the maximum of two locally integrable func-tions is locally integrable.(5) A subset of R is said to be (Lebesgue) measurable if its characteristic func-tion is locally integrable. Show that a countable union of measurable setsis measurable. Hint: Start with two!(6) Define L∞(R) as consisting of the locally integrable functions which arebounded, supR|u| < ∞. If N∞⊂ L∞(R) consists of the bounded functionswhich vanish outside a set of measure zero show that(3) ku + N∞kL∞= infh∈N∞supx∈R|u(x) + h(x)|is a norm on L∞(R) = L∞(R)/N∞.12 RICHARD MELROSE(7) Show that if u ∈ L∞(R) and v ∈ L1(R) then uv ∈ L1(R) and that(4) |Zuv| ≤ kukL∞kvkL1.(8) Show that each u ∈ L2(R) is continuous in the mean in the sense thatTzu(x) = u(x − z) ∈ L2(R) for all z ∈ R and that(5) lim|z|→0Z|Tzu − u|2= 0.(9) If {uj} is a Cauchy sequence in L2(R) show that both (5) and (1) areuniform in j, so given > 0 there exists δ > 0 such that(6)Z|Tzuj− uj|2< ,Z|x|>1/δ|uj|2< ∀ |z| < δ and all j.(10) Construct a sequence in L2(R) for which the uniformity in (6) does nothold.Department of Mathematics, Massachusetts Institute of TechnologyE-mail address:
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