Exercise 2.12

Question

\def	hen{\Rightarrow}
\def\eg{	extit{e.g.} }
\def\ie{	extit{i.e.} }
\def\Def{	riangleq}
\def\R{\usualSet{R}}
\def\C{\usualSet{C}}

Let X be the normed space of all real polynomials in one variable, with %
%
  \begin{align*}
    
orm{}{f}=\int_0^1\lvert f(t)vert\ dt.
\end{align*}
%
Put  %
  $B(f, g)=\int _0^1 f(t)g(t) dt $, %
%
and show that $B$ is a bilinear continuous functional on $X	imes X$ %
which is separately but not continuous.

Cordier · Accepted Answer

\def	hen{\Rightarrow}
\def\eg{	extit{e.g.} }
\def\ie{	extit{i.e.} }
\def\Def{	riangleq}
\def\R{\usualSet{R}}
\def\C{\usualSet{C}}

\begin{proof} Let $f$ denote the first variable, $g$ the second one. %
Remark that %
%
  \begin{align}\label{2_12_2}
    \magnitude{B(f, g)} 
      & < 
orm{}{f} \cdot \max_{[0,1]} \magnitude{g} ; 
  \end{align}
%
which is sufficient (\citeresultFA{1.18}) to assert that any %
%
  $f \mapsto B(f, g)$ 
%
is continuous. The continuity of all %
%
  $g \mapsto B(f,g)$ %
%
follows (Put $C(g, f) = B(f, g)$ and proceed as above). %
Suppose, to reach a contradiction, that $B$ is continuous. %
There so exists a positive $M$ such that, 
%
  \begin{align}\label{2_12_H}
    \magnitude{B(f, g)} \leq M 
orm{}{f}
orm{}{g}.
  \end{align}
%
Put %
%
  \begin{align}\label{2_12_3}
    f_n(X)\Def 2 \sqrt{n}\cdot X^{n}\in \R[X] \quad (\counting{n}), 
  \end{align}
%
so that 
%
  \begin{align}\label{2_12_4}
    
orm{}{f_n} = \frac{2 \sqrt{n}}{n+1} 	endsto{n}{\infty}0.
  \end{align}
%
On the other hand,
%
  \begin{align}\label{2_12_5}
    B(f_n, f_n)= \frac{4 n}{2n+1} > 1.
  \end{align}
%
Finally, we combine %
%
  (ef{2_12_4}) and (ef{2_12_5}) with (ef{2_12_H}) %
%
and so obtain
\begin{align}
 1 < B(f_n, f_n) \leq  M 
orm{}{f_n}^2  	endsto{n}{\infty} 0.
\end{align}
Our continuousness assumption is then contradicted. So ends the proof.
\end{proof}

Exercise 2.12

Answers

Comments

Exercise 2.12

Answers

Comments

Add answer