Let $L^2(\mu)$ and $L^2(\nu)$ with respect to two different positive measures, then they are two Banach spaces. I'm considering whether the space $$L^2(\mu)+L^2(\nu)$$ is still a Banach space?
e.g. $\mu$ be Lebesgue measure, $d\nu=ln(1+|x|)d\mu$, my idea is that since both $L^2(\mu)$ and $L^2(\nu)$ are continuous embedded to the measurable functions space $\mathcal M$, it's done.
If you have two Banach spaces $X,Y$ both continuously embedded in a Hausdorff topological vector space $Z$, you can endow the sum $X+Y$ with the norm $\|z\|=\inf\{\|x\|_X+\|y\|_Y: z=x+y\}$. Then $X+Y$ is a quotient of the Banach space $X\times Y$ with respect to the subspace $X\cap Y$ and hence itself a Banach space.