# Is the sum of such two banach spaces also a banach space?

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.

• What is your definition of $L^2(\mu)+L^2(\nu)$? – Disintegrating By Parts Dec 22 '18 at 20:30
• Are $\mu, \nu$ defined on the same sigma algebra? – Alex Vong Dec 23 '18 at 20:43

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.
• Thank you! But for my case, what's the space $Z$? – Yixuan Zhang Jan 3 '19 at 8:31
• You are right, there is a problem. If your measures are equivalent (i.e, have the same null sets) you can take $Z$ as the space of equivalence classes w.r.t. $\mu$-a.e. equality endowed with convergence in measure. However, if the measures aren't equivalent I don't see a candidate for $Z$ (you need the Hausdorff vector space topology to deduce from the continuity of $(f,g)\mapsto f-g$ that the kernel $X\cap Y$ is closed). – Jochen Jan 3 '19 at 13:33