
\documentclass[psamsfonts]{amsart}

%-------Packages---------
\usepackage{amssymb,amsfonts}
\usepackage[all,arc]{xy}
\usepackage{enumerate}
\usepackage{mathrsfs}
\usepackage{hyperref}

%--------Theorem Environments--------
%theoremstyle{plain} --- default
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{quest}[thm]{Question}

\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{defns}[thm]{Definitions}
\newtheorem{con}[thm]{Construction}
\newtheorem{exmp}[thm]{Example}
\newtheorem{exmps}[thm]{Examples}
\newtheorem{notn}[thm]{Notation}
\newtheorem{notns}[thm]{Notations}
\newtheorem{addm}[thm]{Addendum}
\newtheorem{exer}[thm]{Exercise}

\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}
\newtheorem{rems}[thm]{Remarks}
\newtheorem{warn}[thm]{Warning}
\newtheorem{sch}[thm]{Scholium}

\makeatletter
\let\c@equation\c@thm
\makeatother
\numberwithin{equation}{section}

\bibliographystyle{plain}


\title{Math 6101 - Homework Assignment 3}

\author{Due Thursday, February 9th by 6:00pm}

\date{\today}

\begin{document}



\maketitle


\begin{exer}
Let $X$ be a $\sigma$-compact, locally compact Hausdorff space. Consider $C(X)$ endowed with the topology of uniform convergence on compact sets. Then $C(X)$ is a Fr\'{e}chet space.
\end{exer}

\begin{exer}
Let $X$ be a vector space over $\mathbb K$ and suppose $\| \cdot \|_1$ and $\| \cdot \|_2$ are two complete norms on $X$ such that there exists $C > 0$ with $\| x \|_1 \leq C \| x \|_2$ for all $x \in X$. Show that there exists $C' > 0$ so that $\| x \|_2 \leq C' \| x \|_1$ for all $x \in X$. 
\end{exer}






\begin{exer}\label{exer:openlinearfunctional}
Let $X$ be a topological vector space and suppose $\varphi \in X^*$ is not the zero functional, then $\varphi$ is an open map, i.e., if $G \subset X$ is open then $\varphi(G)$ is also open. 
\end{exer}




\begin{exer}
Let $X$ be a locally convex topological vector space whose topology is defined by a family of seminorms $\mathcal F$. If $\varphi \in X^*$ then there exist $\rho_1, \ldots, \rho_n \in \mathcal F$ and $K > 0$ so that $| \varphi(x) | \leq K \sum_{i = 1}^n \rho_i(x)$, for all $x \in X$.
\end{exer}




\begin{exer}
Consider $L^\infty([0, 1]) = L^1([0, 1])^*$ where $[0, 1]$ is endowed with Lebesgue measure. 
\begin{enumerate}[$($a$)$] 
\item ${\rm span} \{ e^{i 2\pi n t} \}_{n \in \mathbb Z}$ is weak$^*$-dense in $L^\infty([0, 1])$. Hint: Use Lusin's theorem and the Stone-Weierstrass theorem.
\item The sequence $\{ e^{i 2\pi n t} \}_{n \in \mathbb N} \subset L^\infty([0, 1]) = L^1([0, 1])^*$ converges to $0$ in the weak$*$-topology. 
\end{enumerate}
\end{exer}


\begin{exer}
If $X$ is a Fr\'{e}chet space then a subspace $Y \subset X$ is a Fr\'{e}chet space (with respect to the subspace topology) if and only if $Y$ is closed.
\end{exer}


\end{document}









