Polar Factorization Theorem is a result from the foundations of optimal transport. For mathematical objects be a bounded mathematical objectsn() > 0mathematical objectsu L2(, ; n)mathematical objectsu: nmathematical objectsumathematical objectsN nmathematical objects := u\# objects objectss: . It helps organize the relationship between Monge maps, Kantorovich plans, duality, and Wasserstein geometry.