\documentclass[minion]{homework}
\newcommand{\Reals}{\mathbb{R}}

\doclabel{Math F651: Homework 1}
%\docauthor{Your name here.}
\docdate{Due: January 31, 2018}
\begin{document}
\begin{aproblems}

\hproblem R \& Y 2.1

They mean here, show that the putative norm in Example 2.8 really is a norm.

\hproblem R \& Y 2.2

For part b), I know that we have not discussed the Lebesgue space $L^1[0,1]$.  
Don't fret.  Up to a white lie, $L^1[0,1]$ consists of functions 
mapping $[0,1]$ to $\Reals$, and includes the continuous functions.  For
a continous (and hence Riemann integrable function) $f$,
\[
||f||_{L^1[0,1]} = \int_0^1 |f|.
\]
That's enough background to solve the problem.

\hproblem R \& Y 2.3

\hproblem R \& Y 2.4

\hproblem R \& Y 2.5

\hproblem Let $X$ be an normed vector space.  Show that a
sequence $\{x_n\}$ in $X$ converges to $x$ if and only if
\[
\lim_{n\rightarrow\infty} ||x_n-x||=0.
\]

\hproblem Suppose $f$ and $g$ are continuous functions on $[a,b]$.
Show that
\[
\int_a^b fg \le \left[\int_a^bf^2\right]^\frac 12 
\left[\int_a^bg^2\right]^\frac 12.
\]
Then show that if $f$ and $g$ are continuous on $[a,b]$,
\[
||f+g||_2 \le ||f||_2 + ||g||_2
\]
where
\[
||f||_2 = \left[\int_a^b f^2\right]^\frac 12 .
\]


\hproblem R \& Y 2.6

\end{aproblems}
\end{document}

