Exercise 1.6

Question

extit{
Prove that a set $E$ in a topological vector space is bounded if and only if %
every countable subset of E is bounded.
}

Cordier · Accepted Answer

\begin{proof}
It is clear that every subset of a bounded set is bounded. %
Conversely, assume that $E$ is not bounded then pick $V$ %
a neighbourhood of the origin: %
%
No counting number $n=1, 2, \dots$, verifies %
%
  $E\subset nV$ (see Exercise 1 in Chapter 1). %
%
In other words, there exists a sequence %
%
  $\{x_1, \dots, x_n, \dots\} \subset E$ %
%
such that %
%
\begin{align}
  x_n 
otin nV.
\end{align}
%
As a consequence, $x_n /n $ fails to converge to $0$ %
as $n$ tends to $\infty $. %
In contrast, $1/n$ succeeds. %
It then follows from Section 1.30 that %
%
  $\{x_1, \dots, x_n, \dots\}$ %
%
is not bounded. So ends the proof.
\end{proof}
% END

Exercise 1.6

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Exercise 1.6

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