Why is semi-norm special and preferred?

Question

One difference between semi-norm and norm is:

"It is possible for $\|v\| = 0$ for nonzero v, $\|\cdot\|$ being semi-norm"

I see some papers, and they use semi-norm directly.

Why is semi-norm better than norm?

Any simple example or concrete example?

Answer 1

A common use is in topologizing certain function spaces. For example, let $\Omega \subset \mathbb{R}$ be open. We want a topology on $C^\infty(\Omega)$ the collection of smooth functions. Intuitively we'd like convergence $f_n \to f$ to imply the convergence locally of all derivatives. So what we do is we use a family of semi-norms to do it. $\|f\|_{K,n}$ is the sup norm of $f^{(n)}$ over the compact set $K$. There's no sensible reason to "quotient" by polynomials, so we have to admit the possibility that a non-zero function have non-zero $n^{th}$ derivative. (also a smooth function can be zero on some compact subset without being zero).

This becomes important in defining distribution spaces for PDE's