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% MATH 3053: Abstract Algebra I         Fall 2017
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\title{Homework 11} % Replace X with the appropriate number
\author{Analysis} % Do not change this line!
\date{Due: April 23, 2018}
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\begin{enumerate}

\item Let $X,Y$ be normed linear spaces. Prove that if $Y$ is complete the $B(X,Y)$ is a complete space with respect to the operator norm.

\item \textbf{Fourier Series:} With every function $f\in C([0,1])$ we can associate a sequence $a_n$ by
\begin{equation*}
f(x)\mapsto a_n=\int_0^1f(x)\sin(2\pi n x)\,dx.
\end{equation*}
The series $a_n$ is called the Fourier sine series of $g$, and we will denote the map from $C([0,1])$ to sequences by $\mathcal{F}$.
\begin{enumerate}[(a)]
\item Show that $\mathcal{F}$ is a continuous mapping between $(C([0,1]),\|\cdot \|_{L^1})$ and $l^{\infty}$.
\item Show that $\mathcal{F}$ is a continuous mapping between $(C([0,1]),\|\cdot \|_{L^2})$ and $l^{2}$.

\end{enumerate}


\item $l_c$ is the space of all real valued sequences that have only a finite number of non-zero terms.  Recall that $c_0$ is the space of all sequences $a_n\in \mathbb{R}$ satisfying $\lim_{n\rightarrow \infty}a_n=0$. 
\begin{enumerate}[(a)]
\item If $c_0$ is equipped with the $\|\cdot \|_{\infty}$ norm, what is the dual space of $c_0$? That is, what is the space of bounded linear functionals?
\item Show that $l_c$ is a vector space over $\mathbb{R}$.
\item Show that $l_c$ is dense in the sequence space $l^p$ with respect to the $l^p$ norm.
\item Show that the closure of $l_c$ in the sup norm is $c_0$.
\end{enumerate}

\item Show that the mapping defined by
\begin{equation*}
f(x)\mapsto Tf(x)=\int_0^\pi \sin(x-y)f(y)\,dy,
\end{equation*}
maps functions in $C([0,\pi])$ into $C([0,\pi])$. Is this mapping continuous with respect to the $L^1$ and $L^{\infty}$ norms on $C([0,\pi])$, and if so, what are the corresponding induced norms for $T$?

\item Let $\mathcal{A}=\{u\in C^1([a,b]):u(a)=\alpha \text{ and } u(b)=\beta\}$. Consider the functional $I:\mathcal{A}\mapsto  \mathbb{R}$ defined by
\begin{equation*}
I[u]=\int_a^b g(x,u)\sqrt{1+\left(\frac{du}{dx}\right)^2}\,dx,
\end{equation*}
where $g$ is smooth function. 
\begin{enumerate}[(a)]
\item Calculate the \emph{weak form} of the Euler-Lagrange equations for this functional.
\item Calculate the \emph{strong from} of the Euler-Lagrange equations for this functional. 
\end{enumerate}
\item Let $\mathcal{A}=\{u\in C^1([-1,1]):u(-1)=0 \text{ and } u(1)=1\}$. Consider the functional $I:\mathcal{A}\mapsto  \mathbb{R}$ defined by
\begin{equation*}
I[u]=\int_{-1}^1 u(x)^2\left(1-\frac{du}{dx}\right)^2\,dx.
\end{equation*}
Prove that $I$ does not have a minimizer in $\mathcal{A}$. 
\end{enumerate}
\end{document}
