Space of Bounded Functions

Question

Let $B(\mathbb{R},\mathbb{R})$ be the space of all bounded functions on $\mathbb{R}$. Is it possible to define two norms on this space generating two different topologies?

The only norm I know on this space is the supremum norm.

Answer 1

If you are already familiar with the supremum norm, you could introduce weighted variants hereof. For example for any $\alpha<0$ you can define \begin{align*} ||u||_\alpha := \sup_{x\in\mathbb{R}}\, (1+|x|)^\alpha\,|u(x)|. \end{align*} These norms will generate different topologies for different $\alpha$. Consider for example the sequence $u_n(x):=\chi_{[n,n+1]}(x)$, which tends to 0 w.r.t. $||\cdot||_{-1}$ but not w.r.t to $||\cdot||_0$. Consequently, $||\cdot||_{-1}$ and $||\cdot||_0$ generate different topologies.