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{\bf  Final, MAT 324, Fall 2006}
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\begin{enumerate}
\item Give the definition of a norm on a vector space.

\item  Give the definition  of a Cauchy sequence.

\item Give the definition of a complete, normed vector space.

\item Let $C[0,1]$ denote the continuous functions on $[0,1]$ and 
       let $\| f\| = \sup_{[0,1]} |f(x)|$.  Prove that this is a 
       complete, normed vector space. You may quote results about 
       uniform convergence of continuous functuions from analysis.


\item  Prove that $L^2[0,1] \subset L^1[0,1]$.

\item Prove that $L^2[0,\infty)$ is not a subset of $L^1[0,\infty)$.

\item Let $K$ be the closed  subspace of $L^2[0,1]$ which is spanned by the even 
      powers of $x$ (i.e., $1$, $x^2$, $x^4$, \dots). Find the element of $K$ 
      which is closest to $f(x)=x$ in the $L^2$ norm.

\item  Is the sequence $g_n(x) = \chi_{(0,1/n]}(x) \frac 1{\sqrt{x}}$ Cauchy in $L^4$?

\item State H{\"o}lder's inequality.


\item State Minkowski's inequality.

\item Suppose equality holds in Minkowski's inequality, i.e., 
$\|f+g\|_p = \|f\|_p + \|g\|_p$. Prove that $f=ug$ where $|u|=1$
a.e.


\item For $1< p< \infty$, give an example of a function $f\in L^p(0, \infty)$ which is  not in 
$ L^q$ for any $q\ne p$, $q \in (1, \infty)$.



\end{enumerate}

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