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:
Continue reading “On the relationship between L^p spaces and C_c functions for p = infinity”