Hi guys, quick question: In English, how would you say that you're comparing two numbers and treating them as the same if they only differ after a certain decimal point?
@Srivatsan It does seem to be on the right track. Have you tried replacing T by T* throughout and using that T* is injective because T is surjective by hypothesis?
Well, what you're trying to show is equivalent to this. I think the version of the Banach Schauder theorem saying that if T(B_X) dense in B_Y then the induced map X/Ker T -> Y is an isometry and it is onto because T is onto by hypothesis. This should give you a matrix decomposition, but I'm not really in the mood of operator theory.
@Srivatsan You could expand it a little, but I approved it anyway :)
Thanks for your explanation . Apparently, the OP already can make the isometry => T*T = Id connection. All he was looking for is to prove that T* is an isometry.
@Srivatsan Here's a version of that argument. It's the usual proof of the open mapping theorem but pimped a little bit (actually it's the original version of the proof by Banach-Schauder).
@Matt This is very nice to read, thank you! (I liked the end: "Oh, pup needs a stroll, brb"). How did you end up calling her? Aleph-naught, Bet-2 or Gimel-27?
@JonasTeuwen Au début, c'est tellement compliqué, on n'y comprend rien. On peut passer des heures et des heures à chercher sans rien trouver. Dans ces moments-là, mon papa et ma maman sont drôlement inquiets et quand ma maman demande si c'était une bonne idée de faire faire une thèse au petit (c'est moi), mon papa ouvre la bouche sans parler, il agite les bras, et il s'en va lire le journal dans le salon.
Recently they started to serve what is called "Pangasius". A sort of fish that is farmed in large containers in some dirty river in Vietnam and because they have so many fish in one place they give them female hormones and iirc antibiotics so they survive and grow big.
@Srivatsan My honest assessment is: a guy who can't manage to give one decent mathematical answer should refrain from voicing strong opinions on so many textbooks he obviously hasn't digested.
> Now I have put in no effort at all to solve this exercise (except of course reducing it to solving Pell's equation for n=3), which I don't like very much. Unfortunate
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