I am slightly confused about one aspect of the definition of a convergent sequence on a normed vector space. The definition states that a sequence $\{x_n\}$ in the normed space $(X,||\;||)$ is convergent if
$$||x_n-x||\le \varepsilon \;\forall n \gt N$$
where $N$ is some natural number and $x$ is called the limit of convergence. Now every source I have looked at says that $x$ needs to be in $X$ – is this the standard? i.e. would we not call a sequence that converges to an $x \notin X$ convergent?
(A source would be helpful)
Best Answer
$x$ must be an element of $X$ to ensure that $x_n-x$ has meaning and that this is again an element of $X$ so we can look at its norm: $\lVert x_n-x\rVert$
However, if the space $X$ is a subspace of a normed space $Y$ then it is possible for the limit to be outside of $X$. We then say that he sequence doesn't converge in $X$ but it does converge in $Y$.
A subspace $X$ for which every convergent sequence has it's limit still in $X$ is called a closed subspace