In my linear analysis class p-norms were introduced in the context of normed vector spaces and at some point we had to show that the p-norms are ordered. This isn’t difficult to show using induction, but I’d like to emphasise a method I often use, which is to modify an expression into one that lends itself more easily to the tools from analysis.
A p-norm on a finite dimensional vector space over the field of non-negative reals is a mapping defined for as follows:
Now, the key step in showing when is to define
We note that and are smooth non-negative functions of severable variables that have global minima at . More importantly, we have:
From this the general proof follows. I leave it to the reader to work out the details. If you’d like to check your work, my solution is available in the essays section.
I must note that the infinite dimensional case follows easily from the finite-dimensional case by considering divergent series, then convergent series.