Timeline for Theorems for nothing (and the proofs for free)
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 15 at 16:41 | comment | added | Jochen Wengenroth | The title of Banach's and Steinhaus's article from 1927 is Sur le principe de la condensation de singularités points in the opposite direction. For example, there is not only one continuous function on the circle whose Fourier series diverges in one point (du Bois-Reymond) but almost all continuous functions have divergence on a dense set. | |
| Feb 2, 2014 at 12:42 | comment | added | C-star-W-star | @Andrew Stacey: I had similar trouble with understanding how a precompact set doesn't always admit a finite subcover while the larger closure always does - what?! oO - aah adding boundary points is actually good ^^ ...your larger banach space argument reminds me of this =D | |
| Feb 21, 2010 at 11:45 | comment | added | Philip Brooker | @Abhishek Parab: Zabreiko's theorem is proved in Megginson's book 'An Introduction to Banach Space Theory'. It is near the beginning of Section 1.6, which is entitled 'Three Fundamental Theorems'. | |
| Feb 21, 2010 at 2:47 | comment | added | Abhishek Parab | There is a Zabreiko's theorem which extracts the juice of Baire's Category and by invoking it, the Banach-Steinhaus, Open Mapping and Closed Graph theorems come just easily. It says: Every countably subadditive seminorm on a Banach space is continuous. Unfortunately I don't know a good reference. | |
| Nov 13, 2009 at 15:59 | comment | added | Dinakar Muthiah | I agree. It always does seem like you get something for nothing. | |
| Nov 13, 2009 at 15:18 | comment | added | Alex Collins | I agree with Mark: adding stuff tends to rigidify things, think for example of localization. | |
| Nov 13, 2009 at 15:12 | comment | added | Mark Meckes | My intuition (for this kind of issue, anyway) is actually the opposite. If you work with a larger space, then there's more "stuff" that nice things (functions, sequences, whatever) have to play nicely with. So the bigger the space, the nicer they must be. | |
| Nov 13, 2009 at 14:52 | history | answered | Andrew Stacey | CC BY-SA 2.5 |