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Banach's inverse operator theorem

  1. Sep 6, 2014 #1
    Dear friends, I have been trying in vain for a long time to understand the proof given in Kolmogorov and Fomin's of Banach's theorem of the inverse operator. At p. 230 it is said that [itex]M_N[/itex] is dense in [itex]P_0[/itex] because [itex]M_n[/itex] is dense in [itex]P[/itex].

    I am only able to see the proof that [itex](P\cap M_n)-y_0 \subset P_0[/itex] and that [itex](P\cap M_n)-y_0 \subset M_N[/itex] there.
    I obviously realise that [itex]P_0=P-y_0[/itex] and therefore [itex]P_0\cap(M_n-y_0)=(P\cap M_n)-y_0 \subset M_N[/itex], but I don't see why [itex]P_0\subset\overline{M_N}[/itex]...
    What I find most perplexing is that, in order to prove the density of [itex]P_0[/itex] in [itex]M_N[/itex], I would expect something like Let [itex]x[/itex] be such that [itex]x\in P_0[/itex]... then [itex]x\in \overline{M_N} [/itex], while, there, we "start" from [itex]z\in P\cap M_n[/itex] such that [itex]z-y_0\in P_0[/itex], but I don't think that all [itex]x\in P_0[/itex] are such that [itex]x+y_0\in P[/itex]... (further in the proof we look for a [itex]\lambda[/itex] such that [itex]\alpha<\|\lambda y\|<\beta[/itex], i.e. such that [itex]\lambda y\in P_0[/itex])

    Has anyone a better understanding than mine? Thank you very much for any help!!!
     
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  3. Sep 7, 2014 #2

    micromass

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    Hint: ##M_n\subseteq M_N## so ##M_N## is also dense in ##P##. And ##P_0## is an open subset of ##P##.
     
  4. Sep 7, 2014 #3
    Thank you so much!!!
    Forgive me: I don't know how to prove that...
     
  5. Sep 7, 2014 #4

    micromass

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    Indeed, because it's not true. What I meant is that ##P## is an open subset of the sphere ##S##. And ##M_n## (and thus ##M_N##) are dense in there. Apologies for the inconvenience.
     
  6. Sep 7, 2014 #5
    Thank you very much: no problem for any mistyping!!! I see that ##P## is an open subset of the open sphere ##S##, therefore ##P\subset\overline{S}##, and ##M_n## is chosen such that ##S\subset\overline{M_n}##, and ##\overline{M_n} \subset\overline{M_N}## since ##M_n\subset M_N##, so I realise ##M_N## is dense in ##P## (because ##P\subset\overline{S}\subset\overline{M_n}\subset\overline{M_N} ##), but I don't see why ##P_0\subset\overline{M_N}##... :confused:
     
  7. Sep 7, 2014 #6

    micromass

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    ##P_0## is an open subset of the sphere ##S## too.
     
  8. Sep 7, 2014 #7
    Ehm... I cannot see that... :blushing: Of course ##y_0\in M_n\cap S## and ##P\subset S##, and therefore ##P_0\subset S-y_0##, but I see nothing else relevant...
     
  9. Sep 7, 2014 #8

    micromass

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    It's intuitive, no? We know that ##P## is (part of) a sphere inside the sphere ##S##, but it's centered at ##y_0##. Then we translate ##P## to be centered at the origin. I think it should be clear that this translation is still part of ##S##.
     
  10. Sep 7, 2014 #9
    Thank you so much! Is ##S## is centred in ##0##? In that case, yes, I see that ##P_0\subset S##, because if it weren't so, then ##\beta> r## where ##r## is the radius of ##S##, but, in that case, for any ##\varepsilon>0## we could find a ##x\in P## such that ##\|x-y_0\|>\beta-\varepsilon>r-\varepsilon## and, chosing ##\varepsilon=\beta- r##, we would contradict ##P\subset S##, I think. Though, I am not sure how we can chose a ##S## centred in ##0##...
     
  11. Sep 7, 2014 #10

    micromass

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    Haha, ok, ##S## is not necessarily centered in ##0##. Missed that one.

    OK, so ##(4)## basically proves that if ##z\in P\cap M_n##, then ##z-y_0\in P_0\cap M_N##.

    So, to prove ##M_N##is dense in ##P_0##. Take an arbitrary ##x\in P_0##. This is of the form ##x = x^\prime - y_0## with ##x^\prime \in P##. Since ##M_n##is dense in ##P##, we can find a sequence ##(x_n)_n\subseteq M_n\cap P## such that ##x_n\rightarrow x^\prime##. Then obviously by ##(4)##, we have ##(x_n - y_0)_n \subseteq P_0\cap M_N## and ##x_n - y_0\rightarrow x##
     
  12. Sep 7, 2014 #11
    If the book had used a handful of words more...
    I deeply thank you... Now everything is clear.
     
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