You are right now I am editing it so that everything is understood, except that due to a great abuse of notation I put $\mathbb R^\infty$.

Done I edited it.

I know I'm sorry. But I also know that the above answer will get a good upvote, so I don't think it feels a bit unfair. I hope someone can really help me with my question. Greetings.

@RyanBudney Thank you very much for the idea, it is interesting I will try it, if you have time you give it more form as an answer and in case I manage to finish it I will write it too.

Sorry, it was my rush to write. Now I'm going to correct that a bit.

Yes, I did find that article (very interesting, it even puts it as a corollary 5.2) and I also found a thesis based on that article, but I was referring to more current articles (for example after 2000). What I am looking for are new ideas regarding the traditional proofs of this theorem, maybe and just maybe the proof of this theorem has been refined with new tools and in that case I would love to know them.