Given are Banach spaces $B_p\subset C( \mathbb{R})$ with norm $||\cdot||_p = ||g_p \cdot||_\infty$ where $f_p$ is defined via $f_p(x)=(1+|x|^2)^{-p/2}$ and where $||\cdot||_\infty$ is the usual supremum norm. These Banach spaces are defined by
$$B_p:=\lbrace f\in C(\mathbb{R}) : ||f||_p <\infty \rbrace$$
Is it true that
$$B=\bigcap_{p>0} B_p$$
is itself a Banach space for some properly chosen norm?
In a paper I'm reading, $B$ is called an inductive limit. However the definition of inductive limit is a bit different..