- #1
DavideGenoa
- 155
- 5
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 realize 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!
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 realize 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!