Prelude to Wasserstein GAN: More Detailed Proofs

When I was reading this famous paper I found it quite opaque for people as vegetable as I am. Things like “Sobolev norm”, “Kantorovich-Rubinstein duality”, “Pinsker’s inequality” and “Sard’s Lemma” make me feel like an alien who has never learned mathematics on this planet. Besides, some of the proofs are succinct or “is trivial and left as an exercise to the reader”. This article may not aim to explain those intoxicating mathematical terminologies, but to provide more detailed proofs within my ability.