How to show for $1 \leq p < q < r$ the inclusion maps $$L^p \cap L^r \rightarrow L^q$$ $$L^q \rightarrow L^p + L^r$$ are continuous.
where the norms are defined in the following: https://math.stackexchange.com/questions/334628/lp-space-interpolation-norm
Thanks a lot for your help!