I have seen Hölder’s inequality and Minkowski’s inequality proved in several ways but this seems the most perspicuous (to me at any rate).
If and such that
A and satisfying the premise are known as conjugate indices.
Since is convex we have
Substituting in appropriate values gives
Now take exponents.
Let and be conjugate indices with and let and then and
By Young’s inequality
By applying a counting measure to we also obtain
By Hölder’s inequality
and is finite since is a vector space.