Reading time: about 18 minutes (3681 words).

2001 November 26

Instructions. Masters students do any 4 problems. Ph.D. students do any 5 problems. Use a separate sheet of paper for each new problem.

  1. Let $\{f_n\}$ be a sequence of Lebesgue measurable functions on a set $E\subset \mathbb R$, where $E$ is of finite Lebesgue measure. Suppose that there is $M>0$ such that $|f_n(x)|\leq M$ for $n\geq 1$ and for all $x \in E$, and suppose that $\lim_n f_n(x) = f(x)$ for each $x\in E$. Use Egoroff's Theorem to prove that $$\int_E f(x)\ dx = \lim_n \int_E f_n(x)\ dx.$$

  2. Let $f(x)$ be a real-valued Lebesgue integrable function on $[0,1]$.
    a. Prove that if $f>0$ on a set $F\subset [0,1]$ of positive measure, then $\int_F f(x)\ dx > 0.$
    b. Prove that if $\int_0^x f(x)\ dx =0$ for each $x\in [0,1]$, then $f(x)=0$ for almost all $x\in [0,1]$.

  3. State each of the following:
    a. The Stone-Weierstrass theorem
    b. The Lebesgue (dominated) convergence theorem
    c. Holder's inequality
    d. The Riesz representation theorem for $L^p$
    e. The Hahn-Banach theorem.

  1. a. State the Baire category theorem.
    b. Prove the following special case of the uniform boundedness theorem: Let $X$ be a (nonempty) complete metric space and let $F\subseteq C(X)$. Suppose that for each $x\in X$ there is a nonnegative constant $M_x$ such that $$|f(x)| \leq M_x \quad \text{ for all } \quad f\in F.$$ Prove that there is a nonempty \emph{open} set $G\subseteq X$ and a constant $M>0$ such that $|f(x)| \leq M$ holds for all $x\in G$ and for all $f\in F$.

  2. Prove or disprove:
    a. $L^2$ convergence implies pointwise convergence.
    b. $\lim_n \int_0^\infty \frac{\sin(x^n)}{x^n}\ dx = 0$.
    c. Let $\{f_n\}$ be a sequence of measurable functions defined on $[0,\infty)$. If $f_n\rightarrow 0$ uniformly on $[0,\infty)$, as $n\rightarrow \infty$, then $$\varliminf \int_{[0,\infty)} f_n(x)\ dx = \int_{[0,\infty)} \varliminf f_n(x)\ dx.$$

  3. Let $f \colon H\to H$ be a bounded linear functional on a separable Hilbert space H (with inner product denoted by $\langle \cdot, \cdot \rangle$). Prove that there is a unique element $y\in H$ such that $$f(x) = \langle x, y \rangle \quad \text{for all} \quad x\in H \quad \text{and} \quad |f| = |y|.$$ (Hint. You may use the following facts: A separable Hilbert space, $H$, contains a complete orthonormal sequence, $\{\phi_k\}_{k=1}^\infty$, satisfying the following properties:
    i. If $x,y\in H$ and if $\langle x,\phi_k \rangle = \langle y,\phi_k \rangle$ for all $k$, then $x=y$.
    ii. Parseval's equality holds; that is, for all $x\in H$, $\langle x, x \rangle = \sum_{k=1}^\infty a_k^2$, where $a_k = \langle x,\phi_k \rangle$.

  1. Let $X$ be a normed linear space and let $Y$ be a Banach space. Let $B(X,Y) = \{A : A\colon X\to Y \text{ is a bounded linear operator}\}$. Then with the norm $|A| = \sup_{|x|\leq 1} |Ax|$, $B(X,Y)$ is a normed linear space (you need not show this). Prove that $B(X,Y)$ is a Banach space; that is, prove that $B(X,Y)$ is complete.