I understand that POD is about choosing an optimal base, and i have found this Eckart Young theorem
And also have encountered on a book that given an matrix $A$ its projection on POD modes is given by $U^TAU$ where $U$ is given by the SVD decomposition $A=U\Sigma V^T$.
But i don't see the correlation on why should $U$ be the projection.
Ps: This projection is a low-rank structure