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].(adsbygoogle = window.adsbygoogle || []).push({});

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 likeLet [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!!!

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Banach's inverse operator theorem

Loading...

Similar Threads - Banach's inverse operator | Date |
---|---|

Insights Hilbert Spaces And Their Relatives - Part II - Comments | Mar 6, 2018 |

Insights Hilbert Spaces and Their Relatives - Comments | Feb 16, 2018 |

A Hahn-Banach From Systems of Linear Equations | Sep 16, 2017 |

I Inversion of Euler angles | Aug 2, 2016 |

A Convergence of a cosine sequence in Banach space | Jun 4, 2016 |

**Physics Forums - The Fusion of Science and Community**