# Math 224A Fall 2020 Homework 1 ### Updated until Sep 3; due Sep 10. ###### tags: `224a 2020` $\renewcommand{\R}{\mathbb{R}}$ $\newcommand{\C}{\mathbb{C}}$ $\renewcommand{\N}{\mathbb{N}}$ **Clarification.** The purpose of the first three problems is to derive a self-contained proof of both the monotone and dominated convergence using elementary properties of the Riemann integral and Lebesgue measure. The pedagogical reason for doing it this way is that many people may not have seen the Lebesgue integral. If you are familiar with the Lebesgue integral and prefer to do it directly, go ahead, but the proof must be self-contained. 0. Read the proof of Monotone Convergence given the solution of problem (1) in the Lecture 1 notes. 1. Suppose $f_n:\R\rightarrow \R, n\ge 1$ is a sequence of nonnegative functions such that (1) Each $f_n$ is nonincreasing on $\R$, i.e. $f_n(x)\le f_n(y)$ whenever $x\ge y$ (2) $f_n\uparrow f$, i.e., $f_{n+1}(x)\ge f_n(x)$ for every $x\in \R$, converging pointwise as $n\rightarrow\infty$ to $f(x)$. (3) $\int_0^\infty f_n(x)dx \le C$ for all $n$, for some constant $C$. Verify that $$ \lim_{n\rightarrow \infty}\int_0^\infty f_n(x)dx = \int_0^\infty f(x)dx,$$ where the integrals are improper Riemann integrals. 2. Use the monotone convergence theorem to prove *Fatou's Lemma*: If $f_n$ is a sequence of nonnegative functions on $\R$ (not necessarily convergent) then $$\int \lim\inf_{n\rightarrow \infty} f_n(x) dx \le \lim\inf_{n\rightarrow\infty} \int f_n(x)dx.$$ Give an example showing that the nonnegativity hypothesis cannot be removed. 3. Use Fatou's Lemma to prove the Dominated Convergence Theorem (Theorem I.11 of Reed-Simon). (hint: reduce to the nonnegative case first, and show that the necessary $\lim\inf = \lim\sup$ by considering a carefully chosen nonnegative auxiliary function.) 4. Show that the sequence space $\ell_2=\{(x_i)\in \C^{\N}:\sum_i |x_i|^2<\infty\}$ with the dot product is a Hilbert space. 5. Reed and Simon II.2. For part (a) use the definition of the Lebesgue integral from lecture 1 to approximate any $L^2([a,b])$ function by a bounded function. For part (b) you may need to use the fact that the Lebesgue measure is given by $$m(S)=\inf\{\sum_i |I_i|: S\subset \bigcup_i I_i\}$$ where the infimum is taken over $I_i$ a countable collection of open intervals. This proves that $C[a,b]$ is dense in $L^2[a,b]$, which is a part of the proof in Lecture 2 that $L^2[a,b]$ is separable. 6. Reed and Simon II.4. Use part (b) to show that $L^1(\R)$ is not a Hilbert space. 7. Reed and Simon II.6.