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\title{Math 728\\Fall 2004\\Homework 3}
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\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}




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