# Homework 1 ### Updated until Sep 5; due Sep 12 in class. $\renewcommand{\R}{\mathbb{R}}$ 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. (optional) Prove that $L^1$ equipped with the $\|\cdot\|_1$ norm is a complete metric space (Riesz-Fisher Theorem, I.12 of Reed-Simon). The proof is actually given in Reed-Simon so you can either read it or try to do it yourself (I recommend the latter). 5. Prove that the inner product $(x,y)$ in a Hilbert space is jointly continuous in $x,y$ (Richtmyer 1.3.1) 6. Prove that the space $\ell^2$ is complete (Richtmyer 1.4.1-2) 7. Prove that if a Hilbert space has a countable basis then it is separable. 8. Richtmyer 1.7.1-1.7.6. 9. Prove that a linear functional on a hilbert space is bounded if and only if it is continuous. 10. (optional) Reed and Simon II.2. 11. Prove that every inner product space satisfies the *parallelogram identity*: $$\|x+y\|^2 + \|x-y\|^2 = 2\|x\|^2 + 2\|y\|^2.$$ Use this to show that $L^1(\R)$ is *not* a Hilbert space (i.e., its norm does not come from an inner product).