Difference between revisions of "Topological vector space"
From Maths
(Created page with "{{Stub page|msg=Find out if the space must be real, although it looks like it must be. Also find another reference. Demote to B once fleshed out.|grade=A*}} ==Definition== A [...") |
(No difference)
|
Revision as of 17:40, 16 August 2016
Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Find out if the space must be real, although it looks like it must be. Also find another reference. Demote to B once fleshed out.
Contents
Definition
A tuple, [ilmath](V,\mathcal{J})[/ilmath] where [ilmath]V[/ilmath] is a real vector space (a vector space over the field of real numbers, [ilmath]\mathbb{R} [/ilmath]) and [ilmath]\mathcal{J} [/ilmath] is a topology on the set [ilmath]V[/ilmath] is called a topological vector space if[1]:
- The operation of addition is continuous, that is to say that the map [ilmath]\mathcal{A}:V\times V\rightarrow V[/ilmath] given by [ilmath]\mathcal{A}:(u,v)\mapsto u+v[/ilmath] is continuous
- The operation of scalar multiplication is continuous, that is the map [ilmath]\mathcal{M}:\mathbb{R}\times V\rightarrow V[/ilmath] by [ilmath]\mathcal{M}:(\lambda,v)\mapsto \lambda v[/ilmath] is also continuous
Examples
See also
References
| |||||||||||
| |||||||||||||||||||||||
