Timeline for When is the closed unit ball in a smaller Banach space closed in a larger Banach space?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 24, 2015 at 19:20 | vote | accept | Fan Zheng | ||
| Nov 24, 2015 at 15:22 | comment | added | dalry | One condition which follows directly from Hahn-Banach is that there be a family of elements of the dual of $X$ so that the ball of $Y$ is the intersection of the unit balls of the seminorms $|f|$. This suffices to show your result, also in the generalised form that the domain of definition of an (unbounded) self-adjoint operator on a Hilbert space, with its natural structure---also that of a Hilbert space---has your property. Applications: various Sobolev type space obtained by taking powers (not necessarily integral) of the Laplace or Schrödinger operator on Riemann manifolds. | |
| Nov 23, 2015 at 13:17 | comment | added | Jochen Wengenroth | I know the condition as a completeness lemma: If the unit ball of a normed space $X$ is closed in a Banach space $Y$ where $X$ is continuously embedded, then $X$ is also complete. A variant of this easy exercise to locally convex spaces is sometimes attributed to W. Robertson. | |
| Nov 22, 2015 at 20:27 | answer | added | Bill Johnson | timeline score: 6 | |
| Nov 22, 2015 at 9:06 | answer | added | M.González | timeline score: 12 | |
| Nov 22, 2015 at 7:30 | history | asked | Fan Zheng | CC BY-SA 3.0 |