Timeline for Does the uniform boundedness principle holds for multilinear maps as well?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Mar 27 at 23:37 | comment | added | TaQ | In my copy of Rudin's book, on page 43 he precisely writes: "... Banach−Steinhaus theorem (2.5) is often referred to as the uniform boundedness principle," but there is no mention that he himself would follow this convention. | |
| Mar 14 at 21:14 | comment | added | Isaac | I can see in p.44 of the same book that Rudin regards "Banach-Steinhaus" and "uniform boundedness principle" as interchangeable synonyms. | |
| Mar 12 at 14:38 | comment | added | TaQ | (continued) In Rudin's Functional Analysis (p. 43 in my copy) "Banach−Steinhaus" means a certain equicontinuity result. | |
| Mar 12 at 14:38 | comment | added | TaQ | Yes, historically, according to what I know, they obtained the "uniform boundedness principle" for linear maps from a complete normed space to a normed space. However, this basic result has been generalized to varius directions and it has/these have certain rather immediate corollaries. It varies a bit from author to author by which phase one refers to each of these related results. For example, in Horváth's "famous distribution book" on page 216 the phase "Banach−Steinhaus" refers to a result similar to (but less general than) the one in Jarchow's book. | |
| Mar 12 at 13:42 | comment | added | Isaac | My understanding is that uniform boundness principle was established by Banach and Steinhaus and therefore carries their names. Or is there anything I am missing? | |
| Mar 12 at 13:05 | history | answered | TaQ | CC BY-SA 4.0 |