2
$\begingroup$

The following definition is from Janich's Topology book :

Definition (Topological Vector Space). A $\mathbb{R}$-Vector space $(E,\tau)$ with a topological space structure is called a Topological Vector Space of its topological and linear structure are compatible in the following sense :

$A1.$ The subtraction ($-:E^2\rightarrow E$) is continuous.

$A2.$ The multiplication by scalar $\mathbb{R}\times E\rightarrow E$ is continuous.

$A3.$ $(E,\tau)$ is Hausdorff.

Consider definition of "convergence of a sequence to some point", given here.

Again, Let the following definition from wikipedia :

Definition. $\{x_n\}$ is a Cauchy sequence if for each $V\in\tau$ containing $0$, there is some $N$ such that for each $m,n > N $ : $x_m-x_n\in V$.

Now I want to verify that $$\text{Every convergent sequence is cauchy}.$$ But it seems we don't have enough tools to prove.

Let's start ! Suppose $\{x_n\}\rightarrow x$ and let $V$ be a given open set containing $0$. By continuity of addition or subtraction, $V+x$ is a neighborhood of $x$. Therefore, there's a $N$ such that : $$\forall\:m,n>N\:: \left\{ \begin{array}{l}x_m\in V+x\\ x_n\in V+x \end{array}\right.\Longrightarrow \left\{ \begin{array}{l}x_m-x\in V\\ x_n-x\in V \end{array}\right.\tag{*}$$ But by idea of metric spaces, we must choose a "smaller" neighborhood $x$ in order to remove $x$ from the relations $(*)$.

$\endgroup$
  • $\begingroup$ Try to start with:if $x_n$ converges to $x$ then $\forall n>N$ $x_n \in x_N + V$. So $x_m, x_n \in x_N+V$ where $V$ is a neighborhood, now what does this tell you. $\endgroup$ – user108539 Mar 12 '15 at 12:17
  • $\begingroup$ How could you say that for some $N$ : $$\forall\;n>N \:: x_n\in x_N+V$$ ? $\endgroup$ – Fardad Pouran Mar 12 '15 at 13:37
  • $\begingroup$ Definition of convergence $\endgroup$ – user108539 Mar 12 '15 at 13:44
  • $\begingroup$ Can you go to detail ? $\endgroup$ – Fardad Pouran Mar 12 '15 at 13:51
  • $\begingroup$ Can you check this lecture notes page 183 ma.huji.ac.il/~razk/iWeb/My_Site/Teaching_files/TVS.pdf $\endgroup$ – user108539 Mar 12 '15 at 13:56
2
$\begingroup$

You have to consider the continuity of the map $f : E \times E \rightarrow E, (a,b) \mapsto a-b$ at the point $(x,x)$ and choose a neighbourhood $W$ of $x$ such that $f(W,W) \subset V$. Your 'smaller' neighborhood is $W$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.