Reflexivity of intersection of two $L^p$ space

Question

Q:Prove that the linear space $L^{p_1}\cap L^{p_2}$ $(p_1\neq p_2)$ with norm defined by $$\|f\|_{L^{p_1}\cap L^{p_2}}=\|f\|_{L^{p_1}}+\|f\|_{L^{p_2}}$$ is a reflexive Banach space.

I've tried several ways to prove the reflexivity. Such as proving it's a closed subspace of $L^{p_1}$ $(/L^{p_2})$, proving it is a uniformly convex space but all failed. But it seems really complicated to prove directly that it is reflexive. Can someone show a way for me? Thanks for your help!

Answer 1

A possibility is to use a different/equivalent norm on $L^{p_1} \cap L^{p_2}$ which is uniformly convex, e.g. $$\|f\|_{L^{p_1} \cap L^{p_2}}^* = \sqrt{\|f\|_{L^{p_1}}^2 + \|f\|_{L^{p_2}}^2}.$$

Answer 2

Products and closed subspaces are easily seen to be reflexive again and $L^p \cap L^q$ is isomorphic to $\lbrace (f,g)\in L^p \times L^q: f=g\rbrace$. To have this subspace closed you need just some Hausdorff topological space which contains both factors.