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\begin{document}
\title{Problem set 4}
\author{Due date: Mar 26}
\maketitle

\begin{enumerate}
\item Suppose that $X,Y$ are Banach spaces. If $A$ is a bounded linear operator and $\text{ran}(A)$ is second category, then show that $\text{ran}(A)$ is closed.
\item Suppose that $C^{1}[0,1]$, the space of continuously differentiable functions on $[0,1]$ and $C[0,1]$ the space of continuous functions are both equipped with the supremum norm. Suppose that $A:C^{1}[0,1]\to C[0,1]$ be defined by
$Af = f'$, then show that $A$ is unbounded.
\item Suppose $X,Y$ are Banach spaces. Show that there is a constant $c>0$ such that
$\| Ax \| \geq c\|x\|$ if and only if $\cN(A) = \{0\}$ and $\text{ran}(A)$ is closed.
\item If $1<p<\infty$ and $\{x_{n} \} \in \ell^{p}$, then
$\sum_{j=1}^{\infty} x_{n}^{j} y^{j} \to 0$ for every $y\in \ell^{q}$, $1/p+1/q=1$, if and only if $\sup_{n} \| x_{n} \|_{\ell^{p}}<\infty$
and $x_{n}^{j} \to 0$ as $n\to \infty$ for all $j$.
\item If $\{x_{n} \} \in \ell^{1}$, then
$\sum_{j=1}^{\infty} x_{n}^{j} y^{j} \to 0$ for every $y\in c_{0}$, if and only if $\sup_{n} \| x_{n} \|_{\ell^{1}}<\infty$
and $x_{n}^{j} \to 0$ as $n\to \infty$ for all $j$.
\item Suppose $\cH$ is a separable Hilbert space with basis $e_{i}$. Show that a sequence $h_{n} \in \cH$ satifies
$(h_{n},h) \to 0$ for every $h\in\cH$ if and only if $\sup_{n} \| h_{n} \| < \infty$ and $(h_{n},e_{j}) \to 0$
as $n\to \infty$ for each $j$.
\item Suppose that $X$ and $Y$ are Banach spaces, and let $A_{n} : X\to Y$ are pointwise convergent with limit $A:X\to Y$, i.e.
$$
A_{n} f \to Af \quad \forall f\in X \, .
$$
Then the convergence is uniform on compact subsets $U$ of $X$, that is,
$$
\sup_{f\in U} \| A_{n} \phi - A\phi \| \to 0 \quad n \to \infty \,. 
$$
\item A family of bounded operators $\cA = \{A: X\to Y\}$ of linear operators, where $X,Y$ are banach spaces are called
collectively compact, if for each bounded set $U \subset X$, the image set
$\cA(U) = \{ Af: f\in U, A \in \cA\}$ is relatively compact in $Y$. Suppose $X,Y$, and $Z$ are banach spaces. Let
$L_{n}:Y\to Z$ converge pointwise with the limit operator $L:Y\to Z$ (see definition above for pointwise convergence), and let
$\cA$ be a collection of collectively compact operators. Then
$$
\sup_{A\in\cA} \| (L_{n}-L)A \| \to 0\, \quad n\to\infty \, .
$$
\item Suppose that $A_{n}$ is a collection of collectively compact operators which converge pointwise to $A$. Suppose further that $A$ is a compact operator. Assume that $I-A$ is injective. Then for all sufficiently large $n$, for all $n$ with
$$
\| (I-A)^{-1}(A_{n}-A)A_{n} \| < 1 \, ,
$$
the operators $I-A_{n}$ are invertible and uniformly bounded by
$$
\| (I-A_{n})^{-1} \| \leq \frac{1+ \|(I-A)^{-1}A_{n} \|}{1-\|(I-A)^{-1}(A_{n}-A)A_{n} \|} \, ,
$$
For the solutions of the equations
$$
\phi - A\phi = f \quad \text{and} \quad \phi_{n} - A_{n} \phi_{n} f \, ,
$$
there holds the error estimate
$$
\| \phi_{n} - \phi \| \leq \| (I-A)^{-1} \|\frac{\| (A_{n}-A)f \|+\| (A_{n}-A)A_{n}\phi \|}{1-\| (I-A)^{-1}(A_{n}-A)A_{n} \|}
$$
\end{enumerate}
\end{document}