Ricardo A. Sáenz

Due August 28th Problem 1 For $latex n\in\Z_+$, let $latex \mathcal P_n$ the space of polynomials of degree at most $latex n$, seen as functions on $latex [0,1]$. If $latex f_n$ converges uniformly to $latex f$ on $latex [0,1]$, then $latex f\in\mathcal P_n$. Consider the sequence $latex f_n(x) = 1 + \dfrac{1}{2}x + \dfrac{1}{4}x^2 + \ldots + \dfrac{1}{2^n}x^n.$ Then $latex f_n$ converges uniformly in $latex C([0,1])$, but its limit is not a polynomial. Let $latex \mathcal H$ be the subspace of $latex C([0,1])$ of functions satisfying $latex f(1-x) = f(x)$ for any $latex x\in[0,1]$ (these are called  even function on  $latex [0,1]$). Then $latex \mathcal H$ is an infinite dimensional closed subspace of $latex C([0,1])$. Problem 2 Let $latex p$ be a prime number. For $latex r\in\Q$, write $latex r = p^\alpha \dfrac{u}{v}$, where $latex \alpha,u,v\in\Z$ and $latex p$ does not divide neither of $latex u$ nor $latex v$. Define the function $latex |\cdot|_p:\Q\to\Q$ by $latex |r|_p = ...

Due August 21 Problem 1 Let $latex (X,||\cdot||)$ be a normed vector space and $latex x_n, y_n$ sequences in $latex X$ such that $latex x_n\to x$ and $latex y_n \to y$. Then $latex \lambda x_n + \mu y_n \to \lambda x + \mu y$ for any $latex \lambda,\mu\in\mathbb K$. Problem 2 If $latex (X,d)$ and $latex (X,d')$ be homeomorphic metric spaces, then they have the same convergence sequences. However, there exists homeomorphic metric spaces $latex (X,d), (X,d')$ such that only one of them is complete. Problem 3 If $latex (X,||\cdot||)$ and $latex (X,||\cdot||')$ are homeomorphic, then $latex (X,||\cdot||)$ is complete if and only if $latex (X,||\cdot||')$ is complete. Problem 4 Let $latex  f_n$ be the sequence of functions in $latex C([0,1])$ given by $latex \displaystyle f_n(x) = \begin{cases}\sqrt n & 0\le x<\dfrac{1}{n}\\\dfrac{1}{\sqrt x} & \dfrac{1}{n}\le x\le 1.\end{cases}$ Then $latex f_n$ is a Cauchy sequence in $latex (C([0,1]),||\cdot||_1)$ that do...

Due August 14 Problem 1 The function $latex d_T(x,y) = |x^1 - y^1| + \ldots + |x^n - y^n|$ defined for $latex x,y\in\R^n$ is a metric on $latex \R^n$. Problem 2 Two norms $latex ||\cdot||_1$ and $latex ||\cdot||_2$ on a vector space are  equivalent if there exist constants $latex c_1, c_2$ such that $latex c_1||x||_1 \le ||x||_2 \le c_2||x||_1$ for all $latex x\in X$. The norms $latex ||\cdot||_E, ||\cdot||_M$ and $latex ||\cdot||_T$ on $latex \R^n$ are equivalent. If $latex ||\cdot||_1$ and $latex ||\cdot||_2$ are equivalent and $latex B^i_r(x)$ is the ball of radius $latex r$ with center $latex x$ with respect to the metric induced by $latex ||\cdot||_i$, then, for each $latex \e>0$, there exist $latex \delta_1,\delta_2>0$ such that $latex B^1_{\delta_1}(x) \subset B_\e^2(x)$ and $latex B^2_{\delta_2}(x)\subset B_\e^1(x)$. Let $latex ||\cdot||_1$ and $latex ||\cdot||_2$ two norms on $latex X$, and suppose there exist $latex \delta, \e>0$ such that $latex B^1_\delta(0)...