Two norms on $C_b([0,\infty])$ $C_b([0,\infty])$ is the space of all bounded, continuous functions.
Let $||f||_a=(\int_{0}^{\infty}e^{-ax}|f(x)|^2)^{\frac{1}{2}}$
First I want to prove that it is a norm on $C_b([0,\infty])$. The only thing I have problems with is the triangle inequality, I do not know how to simplify 
$||f+g||_a=(\int_{0}^{\infty}e^{-ax}|f(x)+g(x)|^2)^{\frac{1}{2}}$
The second thing I am interested in is how to show that there are constants $C_1,C_2$ such that $||f||_a\le C_1||f||_b$ and $||f||_b\le C_2||f||_a$ so for $a>b>0$ the norms $||.||_a$ and $||.||_b$ are not equivalent.

For the triangle inequality, note  $\|f\|_a=\|e^{-ax/2}f\|_2$, so you can apply Minkowski. 
To see that the norms are not equivalent (and so such constants cannot both exist), let
$$
f_n(x)=\begin{cases}e^{bx/2},&\text{ if } x\in[0,n] \\ \text{line},&\text{ if } x\in[n,n+e^{-bn/2}] \\ 0,&\text{ if }x>n+e^{-bn/2}\end{cases}
$$
Then
$$
\|f_n\|_a\leq\left(\int_0^ne^{(b-a)x}\,dx +1\right)^{1/2}\leq\left(\frac1{b-a}+1\right)^{1/2},
$$
$$
\|f_n\|_b\geq\left(\int_0^n 1\right)^{1/2}=\sqrt n
$$
