6
$\begingroup$

Let's say that $X$ and $Y$ are vector spaces and to be more accurate $Y$ is a dense subspace of $X$. Furthermore we have that $(f_n)_{n\in\mathbb{N}}$ is a sequence in $X^*$, $f\in X^*$ and that $f_n$ converges pointwise to $f$ on $Y$. Can we deduce using continuity and density arguments that $f_n$ converges pointwise to $f$ on $X$?

$\endgroup$
3
  • $\begingroup$ Does $X^*$ denote the set of all continuous linear functionals on $X$? $\endgroup$
    – sranthrop
    Commented Oct 10, 2015 at 9:04
  • $\begingroup$ You can conclude the pointwise convergence on $X$ if the sequence is equicontinuous. $\endgroup$ Commented Oct 10, 2015 at 9:42
  • $\begingroup$ Daniel do u mean equicontinuous on $Y$? $\endgroup$ Commented Oct 10, 2015 at 9:54

1 Answer 1

6
$\begingroup$

Now we cannot. Let $X = c_0$ (the convergent sequences) and $Y = c_{00}$, the subspace of sequences with are zero finally. Define $f_n \in c_0^*$ by $$ f_n(x) = nx_n, $$ then, for any $y \in c_{00}$, we have that $f_n(y) = 0$ finally, hence $f_n|_{c_{00}} \to 0$ pointwise. But, for $x = (1/n) \in c_0$, we have $f_n(x) = 1$ for all $n$, that is $f_n \not\to 0$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .