\documentclass[12pt]{amsart} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} \usepackage[all]{xy} \voffset=-1in \hoffset=-1in \textheight=10in \textwidth=6in \begin{document} \author{} \title{Math 728\\Fall 2004\\Homework 3} \date{} \maketitle \thispagestyle{empty} \pagestyle{empty} \centerline{\it Due Friday September 17, 2004.} \centerline{} \noindent 1. Let $V$ be an inner product space over $\mathbb{R}$. Suppose that $\{e_i\}_{i\in I}$ is an orthonormal basis for $V$ and consider the dual map $\phi_i:V\longrightarrow\mathbb{R}$ given by $\phi_i(e_j)= \begin{cases} 1 & \text{ if } i=j \\ 0 & \text{ if } i\neq j. \end{cases}$ \noindent Finally, let $W\subseteq V^*$ be the subspace of $V*$ spanned by $\{\phi_i\}_{i\in I}$. \begin{itemize} \item[a)] (5 pt) Show that if $\phi\in W$ then there is a unique $u\in V$ such that $\phi(v)=\langle u,v\rangle$ for all $v\in V$. \item[b)] (5 pt) Conclude that if $V$ is finite dimensional, then every linear functional on $V$ (that is, a linear transformation from $V$ to $\mathbb{R}$) is of the form $\langle\circ, u\rangle$ for some (unique) $u\in V$. \end{itemize} \centerline{} \noindent 2. (5 pt) Consider the real vector space $V=\oplus_{i\in I}\mathbb{R}$. Show that the standard dot product" extends to an inner product on this space. \centerline{} \noindent 3. (5 pt) Show that if $V$ is a real inner product space, then $\|v\|=\sqrt{\langle v,v\rangle}$ is a norm on $V$. \centerline{} \noindent 4. (5 pt) Let $\mathfrak{C}$ be the vector space of continuous functions from $[0,1]$ to $\mathbb{R}$. Show that $\langle f,g\rangle=\int_0^1 f(t)g(t)dt$ defines an inner product on $\mathfrak{C}$. \centerline{} \noindent 5. Consider the set of functions $\sin(n\pi x)$, $n\geq 1$ on the interval $[0,1]$. Use the inner product given in problem 4. \begin{itemize} \item[a)] (5 pt) Show that this set of functions is orthogonal. \item[b)] (5 pt) Adjust the set so that it is an orthonormal set. \item[c)] (5 pt) Consider the continous function $f(x)=2x$. For each $n\geq 1$ compute $\langle 2x, \sin(n\pi x)\rangle$ (you have computed the Fourier sine coefficients of the function $f(x)=2x$). \item[d)] (5 pt) Show that the set of continuous functions on $[0,1]$ is infinite dimensional. \end{itemize} \end{document}