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\lhead{\bf Advanced Calculus (MAA 4212)}
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\rhead{Due Wednesday 4/17}
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%\bigskip\noindent\emph{Reminder: Midterm 3 is on Wednesday 4/23.}

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\begin{enumerate}

\Problem We have defined what it means for two norms to be equivalent. Prove that this relation really is an equivalence relation, i.e., that it is symmetric, reflexive, and transitive.

\Problem Consider the vector space $C([0,1])$ with the norms
\[
	\|f\|_2 = \sqrt{\int_0^1 |f|^2 dt}
\]
and
\[
	\|f\|_\infty = \max\{|f(t)|: 0\le t\le 1\}.
\]
Let $f_n(t)=t^n$ (defined on $[0,1]$) and show that
\[
	\lim_{n\to\infty} \|f_n\|_2 = 0
\]
whereas $\|f_n\|_\infty =1$ for all $n$. Conclude that the norms $\|\cdot\|_2$ and $\|\cdot\|_\infty$ are not equivalent. 



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