WJD Open Notebook

2000 November 17

Do as many problems as you can. Complete solutions to five problems would be considered a good performance.

1. a. State the inverse function theorem.
   b. Suppose $L\colon \mathbb R^3 \rightarrow \mathbb R^3$ is an invertible linear map and that $g\colon \mathbb R^3 \rightarrow \mathbb R^3$ has continuous first order partial derivatives and satisfies $|g(x)| \leq C|x|^2$ for some constant $C$ and all $x\in \mathbb R^3$. Here $|x|$ denotes the usual Euclidean norm on $\mathbb R^3$. Prove that $f(x) = L(x) + g(x)$ is locally invertible near $0$.

2. Let $f$ be a differentiable real valued function on the interval $(0,1)$, and suppose the derivative of $f$ is bounded on this interval. Prove the existence of the limit $L = \lim_{x\rightarrow 0^+} f(x)$.

3. Let $f$ and $g$ be Lebesgue integrable functions on $[0,1]$, and let $F$ and $G$ be the integrals $$F(x) = \int_0^x f(t) \ dt, \quad G(x) = \int_0^x g(t) \ dt.$$ Use Fubini's and/or Tonelli's Theorem to prove that $$\int_0^1 F(x)g(x) \ dx = F(1) G(1) - \int_0^1 f(x)G(x) \ dx.$$ Other approaches to this problem are possible, but credit will be given only to solutions based on these theorems.

4. Let $(X, A, \mu)$ be a finite measure space and suppose $\nu$ is a finite measure on $(X, A)$ that is absolutely continuous with respect to $\mu$. Prove that the norm of the Radon-Nikodym derivative $f = \left[\frac{d\nu}{d\mu}\right]$ is the same in $L^\infty (\mu)$ as it is in $L^\infty(\nu)$.

5. Suppose that $\{f_n\}$ is a sequence of Lebesgue measurable functions on $[0,1]$ such that $\lim_{n\rightarrow \infty} \int_0^1 |f_n|\ dx = 0$ and there is an integrable function $g$ on $[0,1]$ such that $|f_n|^2 \leq g$, for each $n$. Prove that $\lim_n \int_0^1 |f_n|^2\ dx =0$.

6. Denote by $\mathcal P_e$ the family of all even polynomials. Thus a polynomial $p$ belongs to $\mathcal P_e$ if and only if $p(x) = \frac{p(x) + p(-x)}{2}$ for all $x$. Determine, with proof, the closure of $\mathcal P_e$ in $L^1[-1,1]$. You may use without proof the fact that continuous functions on $[-1,1]$ are dense in $L^1[-1,1]$.

7. Suppose that $f$ is real valued and integrable with respect to Lebesgue measure $m$ on $\mathbb R$ and that there are real numbers $a<b$ such that $$a \cdot m(U) \leq \int_U f \ dm \leq b \cdot m(U),$$ for all open sets $U$ in $\mathbb R$. Prove that $a \leq f(x) \leq b$ a.e.