There exists definition for continuous spaces with weight?
I am not sure if exist for example definition for $e^{\gamma t }C_{b}(\mathbb R_{+}^n)$ where $C_{b}(\mathbb R_{+}^n)$ is the continuous and bounded space for functions in $\mathbb R^n$. I did not found that definition in literature. ¿Is okay if i define this space as below? For $\gamma<0$
$$e^{\gamma t }C_{b}(\mathbb R_{+}^n):=\{ e^{\gamma t}f(x): f\in C_{b}(\mathbb R_{+}^n) \}$$
with norm $||g||_{e^{\gamma t }C_{b}(\mathbb R_{+}^n)}= ||e^{\gamma t}f||_{C_b}$ where $g(t,x)=e^{\gamma t}f(t,x)$ with $t\geq 0$ and $x\in\mathbb R^{n-1}$?
There is something like this in literature? If somebody knows please i will appreciate any reference. Thank you