Miscellaneous Problems

Let $S \subset \mathbb {R}$ be a set of real numbers with the property that $\left|s_1+\cdots +s_n\right| \leq 1$ for every finite subset $\{ s_1,\ldots ,s_n\} \subset S$. Show that $S$ is countable.

Let $\{ I_{\alpha }\}$, $\alpha \in \Gamma$, be a collection of closed intervals in $\mathbb {R}$. Show that

is countable.

Show that any infinite set of non-empty, mutually disjoint, open sets in a separable metric space $X$ is countable.

Prove that for almost every $x \in [0,2\pi ]$

Let $f \in C(I)$. Show that there exists a sequence of polynomials $\{ p_n\}$ such that $p_n \to f$ uniformly on $I$ and $p_1(x) \leq p_2(x) \leq \cdots$ for every $x \in I$.

Let $f \in C([0,1])$. Show that there is a sequence of odd polynomials $p_n(x)$ with $p_n \to f$ uniformly on $[0,1]$ iff $f(0) = 0$.

Given $f : \mathbb {R} \to \mathbb {R}$ bounded and uniformly continuous and $\{ K_n\}$, $n = 1, 2, 3, \ldots$, $K_n \in L^1$ such that

Let $f : \mathbb {R} \to (-\infty ,\infty )$ be upper semicontinuous and define $m : \mathbb {R} \to [-\infty ,\infty )$ by $m(x) = \varliminf_{y \to x} f(y)$. Let $S = \{ x \mid f(x) - m(x) \geq 1\}$.

Prove that any topological metric space is homeomorphic to a bounded metric space.

Let $M$ be a metric space with distance function $d$, suppose $A : M \to M$ is a distance non-increasing periodic map of order $3$, i.e.,

Let $H$ be a Hilbert space and let $f : H \to \mathbb {R}$ be a continuous convex function such that $f(x_n) \to \infty$ whenever $\left\| x_n\right\| \to \infty$. Prove that $f$ attains a minimum.

$A$ is the subset of $L^1(\mathbb {R})$ consisting of all functions $f$ satisfying $\left|f(x)\right| \leq 1$ a.e.on $\mathbb {R}$. Prove that $A$ is closed in the norm topology of $L^1(\mathbb {R})$.

Let $A$ be a bounded linear operator on Hilbert space $H$. Recall that the adjoint $A^*$ is the unique bounded linear operator on $H$ such that $(Ax,y) = (x,A^*y)$ for all $x, y \in H$.

Suppose that $A, B \subset \mathbb {R}$ are Lebesgue measurable, with $m(A) > 0$, $m(B) > 0$. Show that

contains an interval of positive Lebesgue measure.

Let $X$ be a compact Hausdorff topological space and let $\mu$ be a finite regular Borel measure on $X$. Is it true that if $f : X \to \mathbb {R}$ is $\mu$-measurable then there exists a sequence $\{ f_n\}_{n=1}^{\infty }$ of continuous real-valued functions such that $\lim_{n \to \infty } f_n = f$ a.e.$[\mu ]$? Justify.

Consider a complete metric space $(X,d)$ and a sequence $F_1 \supset F_2 \supset \cdots$ of non-empty and closed subsets of $X$. Assume that for each $n$, the set $F_n$ can be covered by a finite number of balls of radius $1/n$. For each $n$, select a point $x_n \in F_n$. Prove that the sequence $(x_n)_{n=1}^{\infty }$ has a convergent subsequence.

Let $u : \mathbb {R} \to \mathbb {R}$ be arbitrary, and $\chi (x) = \inf_{\left|x-y\right|<1} u(y)$. Show that $\chi$ is upper semi-continuous.

Let $X$ be a Banach space and $Y$ a closed subspace of $X$. Show that every extreme point $g$ of the unit ball of the dual $Y^*$ extends to an extreme point of the unit ball of $X^*$.

Let $f \in L^1[0,1]$, $E_n \subset [0,1]$ be measurable subsets, and

Prove that there exists $\{ n_k\}$ and a continuous function $g(x)$ on $[0,1]$ such that $g_{n_k}(x)$ converges uniformly to $g(x)$ on $[0,1]$.

Prove or disprove: if $K$ and $M$ are metric spaces, $f : K \times M \to \mathbb {R}$ is a continuous function, and $K$ is compact, then the function $g : M \to \mathbb {R}$ defined by

is continuous.