Very quick post on the relationship between , and . I will assume you already know what I am talking about, I’ll just be sharing some intuition on what those mean, but won’t bother with details. It’s more a reminder for me rather than something that intends to be useful, actually, but there’s almost nothing on the Internet about this!
When we discover that (continuous functions with compact support) is dense in , we also discover that it does not hold if and .
What that intuitively means is that if you take away functions in from , you take away something fundamental for : you are somehow taking away a net that keeps the ceiling up.
The fact that it becomes false for limitless spaces () and means that the functions in do not need functions in to survive.
This is reasonable: functions in are not required to exist only in a specific (compact) region of space, whereas functions in do. Functions in are simply bounded – their image keeps below some value, but can go however far they want in x direction. Very roughly speaking, they have a limit on their height, but not on their width.
What we find out, however, is that the following chain of inclusions holds:
That’s reasonable! Think about it:
- Functions in live in a well defined area of space – a confined area of space.
- Functions in are allowed to live everywhere, with the constraint that they become more and more negligible the farther and farther we go. Not required to ever be zero though.
- Functions in are simply required to have an upper bound (a finite one, obviously).
I’m not saying this is simple (advanced analysis is at least as difficult as pitching a nail with a needle as hammer), but after careful thinking, it’s just the way it should be, given the definitions.