Skip to main content Contents Prev Up Next \(\DeclareMathOperator{\RE}{Re}
\DeclareMathOperator{\IM}{Im}
\DeclareMathOperator{\ess}{ess}
\DeclareMathOperator{\intr}{int}
\DeclareMathOperator{\dist}{dist}
\DeclareMathOperator{\dom}{dom}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\cl}{cl}
\DeclareMathOperator{\spn}{span}
\DeclareMathOperator{\proj}{proj}
\DeclareMathOperator\trace{trace}
\DeclareMathOperator\re{\mathrm {Re~}}
\DeclareMathOperator\im{\mathrm {Im~}}
\newcommand\dd{\mathrm d}
\newcommand{\eps}{\varepsilon}
\newcommand{\To}{\longrightarrow}
\newcommand{\hilbert}{\mathcal{H}}
\newcommand{\HH}{\mathcal{H}}
\newcommand{\s}{\mathcal{S}_2}
\newcommand{\A}{\mathcal{A}}
\newcommand\h{\mathcal{H}}
\newcommand{\J}{\mathcal{J}}
\newcommand{\K}{\mathcal{K}}
\newcommand{\M}{\mathcal{M}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathcal{N}}
\renewcommand{\L}{\mathcal{L}}
\newcommand{\T}{\mathbb{T}}
\newcommand{\W}{\mathcal{W}}
\newcommand{\X}{\mathcal{X}}
\newcommand{\D}{\mathbb{D}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\BOP}{\mathbf{B}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\BH}{\mathbf{B}(\mathcal{H})}
\newcommand{\KH}{\mathcal{K}(\mathcal{H})}
\newcommand{\pick}{\mathcal{P}_2}
\newcommand{\schur}{\mathcal{S}_2}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Complex}{\mathbb{C}}
\newcommand{\Field}{\mathbb{F}}
\newcommand{\RPlus}{\Real^{+}}
\newcommand{\Polar}{\mathcal{P}_{\s}}
\newcommand{\Poly}{\mathcal{P}(E)}
\newcommand{\EssD}{\mathcal{D}}
\newcommand{\Lop}{\mathcal{L}(\mathcal{H})}
\newcommand{\cc}[1]{\overline{#1}}
\newcommand{\abs}[1]{\left\vert#1\right\vert}
\newcommand{\set}[1]{\left\{#1\right\}}
\newcommand{\seq}[1]{\left\lt#1\right>}
\newcommand{\norm}[1]{\lVert#1\rVert}
\newcommand{\essnorm}[1]{\norm{#1}_{\ess}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\ran}[1]{\operatorname{ran}#1}
\newcommand{\nt}{\stackrel{\mathrm {nt}}{\to}}
\newcommand{\pnt}{\xrightarrow{pnt}}
\newcommand{\ip}[2]{\left\langle #1, #2 \right\rangle}
\newcommand{\ad}{^\ast}
\newcommand{\inv}{^{-1}}
\newcommand{\adinv}{^{\ast -1}}
\newcommand{\invad}{^{-1 \ast}}
\newcommand\Pick{\mathcal P}
\newcommand\Ha{\mathbb{H}}
\newcommand\Htau{\mathbb{H}(\tau)}
\newcommand{\vp}{\varphi}
\newcommand{\ph}{\varphi}
\newcommand\al{\alpha}
\newcommand\ga{\gamma}
\newcommand\de{\delta}
\newcommand\ep{\varepsilon}
\newcommand\la{\lambda}
\newcommand\up{\upsilon}
\newcommand\si{\sigma}
\newcommand\beq{\begin{equation}}
\newcommand\ds{\displaystyle}
\newcommand\eeq{\end{equation}}
\newcommand\df{\stackrel{\rm def}{=}}
\newcommand\ii{\mathrm i}
\newcommand{\vectwo}[2]
{
\begin{pmatrix} #1 \\ #2 \end{pmatrix}
}
\newcommand{\vecthree}[3]
{
\begin{pmatrix} #1 \\ #2 \\ #3 \end{pmatrix}
}
\newcommand\blue{\color{blue}}
\newcommand\black{\color{black}}
\newcommand\red{\color{red}}
\newcommand\nn{\nonumber}
\newcommand\bbm{\begin{bmatrix}}
\newcommand\ebm{\end{bmatrix}}
\newcommand\bpm{\begin{pmatrix}}
\newcommand\epm{\end{pmatrix}}
\numberwithin{equation}{section}
\newcommand\nin{\noindent}
\newcommand{\nCr}[2]{\,_{#1}C_{#2}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 4.4 Exercises
Exercises Exercises
1. Prove that a closed subspace of a complete metric space is complete with respect to the induced metric. Conclude that
\((c_0, \norm{\cdot}_\infty)\) is a Banach space. (See
Exercise 3.3.6 .)
2. Show that the normed space
\(C(X), \norm{\cdot}_\infty\) of
Checkpoint 3.1.4 is a Banach space.
3. Show that \(RH^2\) is not a Hilbert space.
4.
For which real \(\alpha\) does the function
\begin{equation*}
f_\alpha(t) = t^\alpha e^{-t}, \,\,\, t > 0
\end{equation*}
belong to \(L^2(0,\infty)\text{?}\) When defined, what is \(\norm{f_\alpha}\text{?}\)
5. Prove that the open and closed (unit) balls in a normed space are convex (see
Exercise 3.3.8 ).
6. Prove that the closure of a convex set in a normed space is convex.
7.
Let \(E\) be the Banach space \(\R^2\) with norm
\begin{equation*}
\norm{(x_1, x_2)} = \max\{\abs{x_1}, \abs{x_2}\}
\end{equation*}
Show that \(E\) does not have the closest point property by finding infinitely many points in the closed convex set
\begin{equation*}
A = \{(x_1, x_2): x_1 \geq 1\}
\end{equation*}
which are at minimal distance from the origin.
8. Let \(A\) be a non-empty closed convex set in a Hilbert space. Show that \(A\) contains a unique vector \(a\) of smallest norm that that \(\RE \ip{a}{a-x} \leq 0\) for all \(x \in A\text{.}\)
