Ricardo A. Sáenz

Due October 2nd Problem 1 For any subset $latex A$ of the metric space $latex X$, $latex \diam A = \diam \bar A$. Problem 2 Let $latex A\subset X$ be dense in $latex X$. If $latex E$ is closed in $latex X$ and $latex E\cap A = \emptyset$, then $latex E$ is nowhere dense. Problem 3 If $latex A\subset X$ is a $latex G_\delta$ set and dense in $latex X$, then $latex X\setminus A$ is of the first category. Problem 4 If $latex A$ and $latex X\setminus A$ are dense in the complete space $latex X$, then only one of them can be $latex F_\sigma$ in $latex X$. Problem 5 Let $latex A\subset X$ be countable and dense in the complete space $latex X$ without isolated points. Then $latex A$ is not a $latex G_\delta$ set.

Due September 25th Problem 1 Let $latex f,g\in C(X)$. Then the functions $latex \max(f,g):X\to\R$ and $latex \min(f,g):X\to\R$ are continuous. Problem 2 Let $latex a>0$. The space of continous even functions on $latex [-a,a]$ is a proper closed subalgebra of $latex C([-a,a])$. Problem 3 Let $latex f$ be a continuous function on $latex [a,b]$ such that $latex \displaystyle \int_a^b f(x) x^n dx = 0$ for all $latex n=0,1,2,\ldots$. Then $latex f=0$. Problem 4 Find a nontrivial continuous function $latex f:[0,1]\to\R$ such that $latex \displaystyle \int_0^1 f(x)dx = \int_0^1 f(x) x dx = \int_0^1 f(x)x^2dx = 0.$ Problem 5 Let $latex X,Y$ compact metric spaces, $latex X\times Y$ the product space and $latex \mathcal A$ the algebra generated by the functions $latex f:X\times Y\to \R$ of the form $latex f(x,y) = g(x)h(y), \qquad g\in C(X), \; h\in C(Y).$ Then $latex \mathcal A$ is dense in $latex C(X\times Y)$.

Due September 18th Problem 1 State whether the following families of functions are equicontinuous, pointwise bounded, or both. $latex \{\sin nx\}_{n\ge1}$ in $latex C([0,2\pi])$ $latex \{x^n\}_{n\ge1}$ in $latex C([0,1])$ $latex \{\dfrac{x^n}{n}\}_{n\ge1}$ in $latex C([0,2])$ Problem 2 Let $latex f_n:[a,b]\to\R$ be a monotone sequence of continuous functions, pointwise convergent to the continuous function $latex f:[a,b]\to\R$. Then $latex f_n\rightrightarrows f$ on $latex [a,b]$. Problem 3 Let $latex \Omega\subset\R^m$ be open and $latex (f_n)$ an equicontinuous sequence of functions that converges pointwise in $latex \Omega$. Then $latex (f_n)$ converges uniformly on compact subsets of $latex \Omega$. Problem 4 Let $latex K:[0,1]\times[0,1]\to\R$ be continuous and define the operator $latex L:C([0,1])\to C([0,1])$ by $latex \displaystyle Lf(x) = \int_0^1 K(x,y)f(y) dy.$ Then the closure $latex \overline{L(B_1(0))}$ of the image of the unit ball under $latex L$ is compact in $la

Due September 11th Problem 1 Let $latex f:X\to Y$ be a function, $latex A,B\subset X$ and $latex U,V\subset Y$. $latex f(A\cup B) = f(A)\cup f(B)$. $latex f(A\cap B)\subset f(A)\cap f(B)$. Give an example where $latex f(A\cap B)\not\supset f(A)\cap f(B)$. $latex f^{-1}(U\cup V) = f^{-1}(U)\cup f^{-1}(V)$. $latex f^{-1}(U\cap V) = f^{-1}(U)\cap f^{-1}(V)$. $latex f(f^{-1}(U)) \subset U$. Give an example where $latex f(f^{-1}(U))\not\supset U$. $latex f^{-1}(f(A)) \supset A$. Give an example where $latex f^{-1}(f(A))\not\subset A$. Problem 2 If $latex X$ is sequentially compact and $latex f:X\to Y$ is continuous, then $latex f(X)$ is sequentially compact. Prove it directly using the definition of sequential compactness. Problem 3 Give a set $latex X$ and two metrics $latex d,d'$ on $latex X$ such that $latex (X,d)$ and $latex (X,d')$ are homeomorphic, but $latex f:X\to X$ given by $latex f(x)=x$ is not uniformly continuous. Problem 4 1.  Let $latex \mathcal I:C([0,1]\

Due September 4th Problem 1 Let $latex X$ be sequentially compact. Then every infinite subset of $latex X$ has a limit point. (Do not use the Bolzano-Weierstrass theorem.) Problem 2 Let $latex E$ be a compact subset of $latex \R$. Then it has a minimum and a maximum. Problem 3 Let $latex A$ be a bounded infinite subset of $latex \R^l$. Then it has a limit point. Problem 4 Let $latex (x_n)$ be a bounded sequence in $latex \R^l$. Then is has a convergent subsequence. Problem 5 Let $latex A$ be a nonempty set in the metric space $latex (X,d)$ and, for $latex \e>0$, define $latex A_\e = \{x\in X: d(x,A) < \e\}.$ Then $latex A_\e$ is open in $latex X$.