\documentclass[12pt]{article}
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\def\Cx{\mathbb{C}}
\def\Nl{\mathbb{N}}
\def\Ra{\mathbb{Q}}
\def\Rl{\mathbb{R}}
\def\Ts{\mathbb{T}}
\def\Ir{\mathbb{Z}}

\begin{document}

\centerline{\textbf{Problem Set 6}}
\centerline{\textbf{Math 201A}: \textbf{Fall 2016}}

\bigskip
\bigskip
\noindent
\textbf{Problem 1.} Is $(x^n)_{n=0}^\infty$ a Schauder basis of $C([0,1])$?



\bigskip\noindent
\textbf{Problem 2.} Let $A : \Rl^n \to \Rl^m$ be a linear map with matrix $(a_{ij})$, so $y=Ax$
if $y _i = \sum_{j=1}^n a_{ij} x_j$. Equip $\Rl^n$ and $\Rl^m$ with the one-norms
$\|x\|_1 = \sum_{j=1}^n |x_j|$, $\|y\|_1 = \sum_{i=1}^m |y_i|$. Show that the corresponding
operator norm of $A$ is given by the maximum absolute column sum
\[
\|A\|_1 = \max_{1\le j \le n} \sum_{i=1}^m |a_{ij}|.
\]



\bigskip
\noindent
\textbf{Problem 3.} Suppose that $(X, \|\cdot \|)$ be a Banach space and $M \subset X$ is a closed linear subspace.
Let $X/M = \left\{x+M : x\in X\right\}$ denote the the quotient space of $X$ by $M$, where
\[
x+M = \left\{y\in X : \text{$y=x+m$ for some $m\in M$}\right\},
\]
and define $\|x+M\| = \inf_{m\in M} \|x+m\|$.
Show that $\|\cdot\| : X/M \to \Rl$ defines a norm on $X/M$ and $X/M$ is a Banach space
with respect to this norm.


\bigskip
\noindent
\textbf{Problem 4.} (a) Suppose that $V$ is a proper closed subspace of a normed linear space
$X$ and $0<r<1$. Show that there exists $x\in X$ such that $\|x\|=1$ and $d(x,V) =
\inf_{v\in V} d(x,v) > r$. \textsc{Hint.} Choose $x_1\in X\setminus V$. If $R = d(x_1,V)$,
choose $v_1\in V$ such that $d(x_1,v_1) < R +\epsilon$, and let $x= (x_1-v_1)/\|x_1-v_1\|$.

\smallskip\noindent
(b) Show that the closed unit ball $\{x\in X : \|x\|\le 1\}$ of an infinite dimensional
Banach space $X$ is not compact.


\bigskip
\noindent
\textbf{Problem 5.}  Let $c$ be the
Banach space of convergent real sequences and let $c_0\subset c$ be the Banach space of real sequences
that converge to $0$, both equipped with the
sup-norm $\|(x_n)\| = \sup_{n\in \Nl} |x_n|$.

\smallskip
\noindent(a) Show that $c$ and $c_0$ are homeomorphic. \textsc{Hint.} Define $\Phi : c \to c_0$ by
\[
\Phi(x_1, x_2,x_3,x_4\dots) = (a,x_1-a,x_2-a,x_3-a,\dots),\qquad a = \lim_{n\to\infty} x_n.
\]
(b) For any $x\in c_0$ with $\|x\|=1$, show that there exist $y,z\in c_0$
such that $y\ne z$, $\|y\|=\|z\|=1$, and $x = \frac{1}{2}(y+z)$.
Deduce that $c_0$ and $c$ are not isometrically isomorphic.



\end{document}
