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\title{MATH 202A \\ Applied Algebra I \\ Fall 2019}

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\section*{Homework week 8}
\vspace{0.5\baselineskip}

Due by the beginning of class on Friday 22nd November (hand in via Gradescope).
\vspace{0.5\baselineskip}

For two norms $\|\cdot\|$, $\|\cdot\|'$ on the same vector space $V$, we write $\|\cdot\| \le \|\cdot\|'$ to mean $\|v\| \le \|v\|'$ for all $v \in V$.

Throughout, you may use any form on the Hahn--Banach theorem that has been stated in lectures (proven or otherwise).

\begin{enumerate}[label={\bf\arabic*.}, itemsep=0.8\baselineskip, leftmargin=2em, labelsep=2em, listparindent=0em, parsep=0.3\baselineskip]

  \item For each of the following pairs of norms $\|\cdot\| , \|\cdot\|'$ on $\CC^n$, find, with proof, the smallest constant $C>0$ and the largest constant $c>0$ such that $c \|v\| \le \|v\|' \le C \|v\|$.  (These constants may depend on $n$.)
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item $\|\cdot\| = \|\cdot\|_2$, $\|\cdot\|' = \|\cdot\|_\infty$.
      \item $\|\cdot\| = \|\cdot\|_2$, $\|\cdot\|' = \|\cdot\|_1$.
      \item $\|\cdot\| = \|\cdot\|_7$, $\|\cdot\|' = \|\cdot\|_{17}$.
    \end{enumerate}
    [If you prefer to prove a general result, feel free.]

  \item Let $V$ be a finite-dimensional vector space.
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item Suppose $\|\cdot\|$, $\|\cdot\|'$ are two norms on $V$ such that  $\|\cdot\| \le \|\cdot\|'$.  Prove that ${\|\cdot\|'}^\ast \le \|\cdot\|^\ast$.

      \item If $\|\cdot\|$ is a norm on $V$ and $\alpha \in \RR_{>0}$ is a scalar, write $\alpha\|\cdot\|$ for the norm $v \mapsto \alpha \|v\|$.  Prove that the dual norm to $\alpha \|\cdot\|$ is $(1/\alpha) \|\cdot\|^\ast$.
    \end{enumerate}

  \item Let $V$, $W$ be finite-dimensional normed spaces, with the norms denoted $\|\cdot\|_V$, $\|\cdot\|_W$.  Consider $V^\ast$, $W^\ast$ as normed spaces with the norms $\|\cdot\|_V^\ast$, $\|\cdot\|_W^\ast$ respectively.

    Also, suppose $\phi \colon V \to W$ is a linear map.
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item Prove that $\|\phi^\ast\|_{\op} \le \|\phi\|_{\op}$.
      \item Prove that $\|\phi^\ast\|_{\op} = \|\phi\|_{\op}$.

        \noindent [{\bf Hint}: stating part (a) before part (b) is a hint.]

      \item Reflect on the relationship between your proof of Pset 4 Q3 on the one hand, and (a) here together with Example 5.3.1 on the other hand.  [You do not need to write anything for this part.]
    \end{enumerate}

  \item Let $V$ be a finite-dimensional vector space and $\|\cdot\|$, $\|\cdot\|'$ two norms on $V$.  Consider the following functions $V \to \RR$:---
    \begin{align*}
      v &\mapsto \|v\| + \|v\|' \\
      v &\mapsto \max(\|v\| , \|v\|') \\
      v &\mapsto \inf_{\substack{x,y \in V \\ x+y=v}} \big( \|x\| + \|y\|' \big) \\
      v &\mapsto \inf_{\substack{x,y \in V \\ x+y=v}} \max(\|x\| , \|y\|').
    \end{align*}
    \def\coplus{\operatorname{coplus}}
    \def\comax{\operatorname{comax}}
    We denote there respectively by $\|\cdot\| + \|\cdot\|'$, $\max(\|\cdot\|, \|\cdot\|')$, $\coplus(\|\cdot\|, \|\cdot\|')$ and $\comax(\|\cdot\|, \|\cdot\|')$.  Note these last two are non-standard notation.

    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item Very briefly, verify that these are all norms on $V$.
      \item Prove that $\big(\|\cdot\| + \|\cdot\|'\big)^\ast \le \comax(\|\cdot\|^\ast, {\|\cdot\|'}^\ast)$.
      \item Prove that $\big(\max(\|\cdot\| , \|\cdot\|')\big)^\ast \le \coplus(\|\cdot\|^\ast, {\|\cdot\|'}^\ast))$.

        \noindent [Note: these proofs may have many common steps.  You should feel free to merge you answers or otherwise avoid repeating yourself too much.]

      \item[\bf (d')] Prove that $\big(\comax(\|\cdot\| , \|\cdot\|')\big)^\ast \ge \|\cdot\|^\ast + {\|\cdot\|'}^\ast$.

        [{\bf Hint:} Given $\phi \in V^\ast$, by definition you are trying to find $v \in V$ such that $|\phi(v)|$ is large but $ \|v\|_{\comax}$ is small.  Note this is the same as trying to find $x,y \in V$ with $|\phi(x+y)|$ large but $\max(\|x\|,\|y\|')$ small.  You can do this directly from the definitions of $\|\phi\|^\ast$ and ${\|\phi\|'}^\ast$.]

      \item[\bf (e')] Prove that $\big(\coplus(\|\cdot\| , \|\cdot\|')\big)^\ast \ge \max(\|\cdot\|^\ast, {\|\cdot\|'}^\ast)$.

        [{\bf Hint:} see (d').]

      \item[\bf (f)] Prove that in fact equality holds in (b),(c),(d'),(e').

        [{\bf Hint:} now you shouldn't have to do more work.]
    \end{enumerate}

  \item Consider the vector space $\CC^n$ with the usual $\ell^p$-norms.
    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.3\baselineskip, parsep=0.3\baselineskip, listparindent=0em]
      \item Prove that $\|\cdot\|_2 \le \frac12 \|\cdot\|_{4/3} + \frac12 \|\cdot\|_4$.

        [Here we use the notation developed in previous questions.]

      \item Let $x \in \CC^n$ be a vector with $\|x\|_2 \le 1$.  Prove that there is a decomposition $x = y+z$ where $y,z \in \CC^n$ satisfy $\|y\|_{4/3} \le 1/2$ and $\|z\|_4 \le 1/2$.

        [{\bf Note}: you could give a direct proof of this statement, i.e.\ by finding an explicit formula for $y_i$ and $z_i$, using calculus or otherwise.  This is strongly discouraged.  You should attempt to obtain this fact ``for free'' from (a), by leveraging the results from the rest of the pset.]
    \end{enumerate}
\end{enumerate}
\end{document}
