Talk:Examples of vector spaces: Difference between revisions

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Fields: that link is somewhat out of place
Function spaces and generalized coordinate spaces
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:The [[vector space]] article already discusses [[module (mathematics)|module]]s in the section on generalizations. The link is somewhat out of place here. If there were an article on [[examples of modules]], it might be appropriate to link it from here (I'm not claiming such an article should exist). -- [[User:Fropuff|Fropuff]] 02:41, 12 January 2006 (UTC)
 
== Function spaces and generalized coordinate spaces ==
 
In this article, the notation '''F'''<sup>''X''</sup> is used for the set of maps from ''X'' to the field '''F''' with finitely many nonzero terms (aka, the generalized coordinate space with basis isomorphic to ''X''). I believe this is a bad idea: the notation ''Y''<sup>''X''</sup> is standard in set theory for the set of '''''all''''' maps from ''X'' to ''Y'', and gives rise to the notion of an [[exponential object]] in category theory. In fact, in this article, the notation '''F'''<sup>'''N'''</sup> is already mentioned as a notation for the set of all maps from the natural numbers to '''F'''.
 
I think the notation ''V''<sup>''X''</sup> should be given as a notation for the function space of all maps from ''X'' to ''V'', then specialized to the case ''V''='''F'''. Then the generalized coordinate space should be defined as a subspace of '''F'''<sup>''X''</sup>, with a different notation. I guess the correct notation would be to use the direct sum <math>\bigoplus_X \mathbf F</math> or coproduct. Comments? [[User:Geometry guy|Geometry guy]] 15:55, 12 February 2007 (UTC)