Norms in normed spaces are always assumed to be finite? True? But where are infinite norms then?
This is the "background" that I have been presented as to why questioning the finiteness of Cauchy-Schwarz etc. in normed spaces is unreasonable. Because $\mathbb{R}$ doesn't have $\pm \infty$. That while one may construct infinite sums inside norms, then in normed spaces they're not assumed to exist.
Where do then, however, infinite norms exist?