Homework 1

Updated until Sep 5; due Sep 12 in class.

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1. Suppose
   $f_n:\mathbb{R}\to \mathbb{R},n\ge 1$ is a sequence of nonnegative functions such that
   (1) Each
   $f_n$ is nonincreasing on
   $\mathbb{R}$, 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
   $x\in \mathbb{R}$, converging pointwise as
   $n\to \mathrm{\infty }$ to
   $f(x)$.
   (3)
   ${\int }_0^{\mathrm{\infty }}{f}_n(x)dx\le C$ for all
   $n$, for some constant
   $C$.
   Verify that
   $$\underset{n\to \mathrm{\infty }}{lim}{\int }_0^{\mathrm{\infty }}{f}_n(x)dx={\int }_0^{\mathrm{\infty }}f(x)dx,$$ 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
   $\mathbb{R}$ (not necessarily convergent) then
   
   $$\int lim\underset{n\to \mathrm{\infty }}{inf}{f}_n(x)dx\le lim\underset{n\to \mathrm{\infty }}{inf}\int f_n(x)dx.$$
   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
   $liminf=limsup$ by considering a carefully chosen nonnegative auxiliary function.)
4. (optional) Prove that
   $L^1$ equipped with the
   $\Vert \cdot {\Vert }_1$ 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:
    
    $$\Vert x+y{\Vert }^2+\Vert x-y{\Vert }^2=2\Vert x{\Vert }^2+2\Vert y{\Vert }^2.$$
    Use this to show that
    $L^1(\mathbb{R})$ is not a Hilbert space (i.e., its norm does not come from an inner product).