1
$\begingroup$

My question might be a little vague, but is there a way to know which type of convergence (i.e pointwise, uniform) to use when proving that a subspace is dense in a certain space.

For example if we were to consider the space of twice continuously differentiable and bounded functions on $\mathbb{R}$ and wanted to prove density of a certain subspace with additional constraints in that space, how would we know whether to use pointwise or uniform convergence?

As pointwise is much weaker, could it therefore be that density of the subspace holds for the weak topology, but not for the strong one?

$\endgroup$
  • 1
    $\begingroup$ A set $S\subset X$ is dense if $\overline S=X$. The closure of course only makes sense with respect to a given topology. So it seems you've answered your own question. $\endgroup$ – Math1000 Aug 3 '15 at 17:12
  • 1
    $\begingroup$ If you want to show a set is dense on one topology, use that topology. On the other and completely different hand, if you want to show a set is dense in another topology you should use that topology. $\endgroup$ – David C. Ullrich Aug 3 '15 at 17:15
1
$\begingroup$

To speak of density of a subspace $Y$ in a space $X$, $X$ and $Y$ have to be topological spaces. So, the answer to your question is: it depends on the topology of $X$. In your example, you may equip $X$ either with the topology of uniform convergence or the topology of pointwise convergence.

$\endgroup$
  • $\begingroup$ Thanks Nocturne. Just one thing, why do you specify uniform convergence on compact sets instead of just uniform convergence for this example? $\endgroup$ – user223935 Aug 3 '15 at 21:35
  • $\begingroup$ @user223935 I didn't notice your functions were bounded. The topology of uniform convergence in your example would be the topology induced by the metric $d(f,g)=\sup\limits_{x \in \mathbb{R}} |f(x)-g(x)|$ which is well-defined. $\endgroup$ – Nocturne Aug 3 '15 at 21:50

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.