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\title{ HW III for Math 6210\\ }
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\noindent 
1)  Let $V$ be a normed space and $U$ a closed subspace. For every $x\in V$, let 
\[ || x+ U|| = \inf \{||x+y|| ~|~ y\in U\}.\]  
\begin{enumerate} 
\item Prove that $||x+U||$ is a norm on $V/U$. 
\item For every $\epsilon>0$ there exists $x\in V$ such that $||x||=1$ and $||x+U||\geq 1-\epsilon$. 
\item If $V$ is complete, so is $V/U$.  
\item Let $P: V \rightarrow V/U$ be $P(x)=x+U$. Prove that $P$ has norm 1. 
\end{enumerate} 

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\noindent 
2) Let $T : V \rightarrow W$ be a bounded map. Let $U$ be the kernel of $T$. 
\begin{enumerate} 
\item Prove that $U$ is closed. 
\item Prove that $||S||=||T||$ where $S: V/U \rightarrow W$ such that $S(x+U) = T(x)$, for all $x\in V$. 
\end{enumerate} 


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\noindent 
3) Let $T : V \rightarrow W$ be a map between two normed spaces, where $W$ is finite dimensional. Let $U$ be the kernel of $T$. Prove that $T$ is bounded if 
$U$ is closed. 

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\noindent 
4)  Let $(X,\mathcal M, \mu)$ be a measure space, and $E\in \mathcal M$ of finite and positive measure.
 Let $T : L^1(X) \rightarrow \mathbb R$ be defined by  $T(f)=\int_E f$. Prove that $T$ is a bounded functional, and compute its norm. 


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\noindent 
5)  Let $T : V \rightarrow U$ be a bounded map between two normed spaces.  Let $T^* : U^* \rightarrow V^*$ be defined by $T^*(f) = f\circ T$ for all $f\in U^*$ (the adjoint map). 
\begin{enumerate} 
\item  Prove that $||T^*||= ||T||$. 
\item If we identify $V$ and $U$ with their canonical images in $V^{**}$ and $U^{**}$ prove that the restriction of $T^{**}$ to $V$ coincides with $T$. 
\end{enumerate} 

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\noindent 
6) Let $V=C([0,1])$ be the space of continuous functions equipped with the sup norm. Let $U=C^1([0,1])$ be the subspace of continuously differentiable functions. 
Let $T: U \rightarrow V$ be defined by $T(f)=f'$. Prove that the graph of $T$ is closed, and that $T$ is not bounded. 
 

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\noindent 
7)  Let $T : V \rightarrow U$ be a linear map between two Banach spaces such that if $f$ is a continuous functional  on $U$, then $f\circ T$ is a continuous functional on $V$. 
Prove that $T$ is bounded. Hint: prove that the graph of $T$ is closed, to that end use Theorem 5.8 c on page 159. 

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\noindent 
6) (More magic.)  Let $V$ be a space complete with respect to two norms $||\cdot ||_1 \leq ||\cdot ||_2$. Prove that the norms are equivalent. 
 





