Timeline for inequivalent norms
Current License: CC BY-SA 4.0
3 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 29, 2020 at 1:08 | vote | accept | user92646 | ||
| Apr 28, 2020 at 22:56 | comment | added | YCor | Note: proving that there's a non-equivalent norm is easier than all this (no need to use the $2^{\aleph_0}$ fact). Namely, if $X$ is an infinite-dim normed space, find a basis $(v_i)_{i\in I}$ contained in the 1-sphere. For every family $(t_i)$ of nonzero scalars, $v_i\mapsto t_iv_i$ defines a linear automorphism. Choosing $(t_i)$ unbounded yields a non-continuous automorphism. Choosing different families easily yields a group of $c$ such automorphisms, each unbounded except identity, and hence this yields $c$ pairwise non-equivalent norms (but $2^c$ is hopeless if the dimension is countable). | |
| Apr 28, 2020 at 13:00 | history | answered | YCor | CC BY-SA 4.0 |