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\title{Math studies algebra Spring 2018: Homework 6}

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\author{JC}

\begin{document}

\maketitle

   This homework is due by class time on Monday 2 April. It must be typeset
   (preferably in\ \LaTeX) and submitted as a PDF file on the Canvas site,
   with a filename of the form \begin{verbatim} andrewID_alg_homeworknumber.pdf \end{verbatim}   
   For each minute that it is late, the grade will be reduced by 10 percent. 

\medskip

\medskip

\begin{enumerate}

\item  Prove that if $n$ is finite, every linear map from ${\mathbb R}^n$
  to $\mathbb R$ (both considered as metric spaces with the usual Euclidean metric) 
  is continuous.

\item  (Tricky, ask me if you want a hint) Prove that there is a linear map from
  $l^2(\mathbb R)$ to $\mathbb R$ which is not continuous.

\item  Let $V$ be a FD real IP space and let $T: V \rightarrow V$ be a linear map such that
  $T$ respects the IP, that is $Tv \cdot Tw = v \cdot w$ for all $v, w \in V$. Prove that
  $T$ is invertible. Will the conclusion remain true in general if we drop the hypothesis that
  $V$ is FD?

\item Let $V$ be a real vector space and let $W = V \oplus V$. Define a function from
  ${\mathbb C} \times W$ to $W$ by $(r + i s)(v_1, v_2) = (r v_1 - s v_2, s v_1 + r v_2)$
  for $r, s \in {\mathbb R}$ and $v_1, v_2 \in V$. Prove that this definition makes $W$ into
  a complex vector space. If $dim(V) = n$ then find the dimension of $W$ considered as
  a real space and considered as a complex space.

\item  In last week's homework you showed that if $R$ is a subring of $S$ and $M$ is an $R$-module,
  then we can view $S \otimes_R M$ as an $S$-module. Show that the construction of Q4 above can be viewed
  in this light by thinking of $W$ as ${\mathbb C} \otimes_{\mathbb R} V$.

\item Let $V$ be a real IP space and let $W$ be the complex vector space defined as in Q4.
  Show that we can define a complex IP on $W$ which extends the real IP on $V$ (where we
  view $V$ as embedded in $W$ via the map $v \mapsto (v, 0)$). 
  
  
   
\end{enumerate}  






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