MATH 447 Homework 10: Norms and Density in Lp Spaces - Prof. Roger Smith, Assignments of Mathematics

Download MATH 447 Homework 10: Norms and Density in Lp Spaces - Prof. Roger Smith and more Assignments Mathematics in PDF only on Docsity! MATH 447, HOMEWORK 10, DUE APR 24 Q1. If f is bounded and measurable on [a, b], let ‖f‖∞ = inf{c ≥ 0 : |f(x)| ≤ c a.e.}. Prove that this is a norm, and that lim p→∞ ‖f‖p = ‖f‖∞. Q2. Prove that simple functions, step functions and continuous functions of compact support are all dense in Lp(R) for each p ∈ (1,∞). Q3. If f ∈ L1(R), prove that lim t→0 ∫ R |f(x)− f(x+ t)| dx = 0. (Try step functions first) Q4. If 1 < p1 < p2 <∞, find f and g so that f ∈ Lp1(R), f /∈ Lp2(R), g ∈ Lp2(R), g /∈ Lp1(R). Q5. Let H be a Hilbert space with an orthonormal basis {xn}n≥1. Prove that ∑ αnxn converges if and only if ∑ α2n converges. 1