Okay I took the nerd bait and signed up for LW to say:
For your example to work you need to restrict the domain of your functions to some compact e.g. C1([0,1])because the uniform norm requires the functions to be bounded.
Also note this example works because you’re not using the “usual” topology on C1([0,1]) which also includes the uniform norm of the derivative and makes the space complete. It is much more difficult if the domain is complete!
For your example to work you need to restrict the domain of your functions to some compact e.g. C1([0,1])because the uniform norm requires the functions to be bounded.
Hmm, is this really a substantive problem? Call it an “extended norm” instead of a norm and everything in the post works, right? My reasoning: An extended norm yields an extended metric space, which still generates a topology — it’s just that points which are infinitely far apart are in different connected components. Since you get a perfectly valid topology, it makes perfect sense for the post to talk about continuity. Or at least I think so; am I missing something?
Okay I took the nerd bait and signed up for LW to say:
For your example to work you need to restrict the domain of your functions to some compact e.g. C1([0,1])because the uniform norm requires the functions to be bounded.
Also note this example works because you’re not using the “usual” topology on C1([0,1]) which also includes the uniform norm of the derivative and makes the space complete. It is much more difficult if the domain is complete!
Hmm, is this really a substantive problem? Call it an “extended norm” instead of a norm and everything in the post works, right? My reasoning: An extended norm yields an extended metric space, which still generates a topology — it’s just that points which are infinitely far apart are in different connected components. Since you get a perfectly valid topology, it makes perfect sense for the post to talk about continuity. Or at least I think so; am I missing something?
Perhaps! I’m not familiar with extended norms. But when you say “let’s put the uniform norm on C1(R)” warning bells start going off in my head 😅
Thanks for commenting—and for your patience. I’ve changed the domain to be an arbitrary closed interval and credited you at the bottom.