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Topic: F^I isomorphic to finite(F^I)
Replies: 11   Last Post: Mar 8, 2013 3:45 PM

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Robin Chapman

Posts: 379
Registered: 5/29/08
Re: F^I isomorphic to finite(F^I)
Posted: Mar 6, 2013 6:30 AM
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On 05/03/2013 23:03, Kaba wrote:
> Hi,
> Let F be a field, and I be a set. Denote by finite(F^I) the set of
> functions from I to F which are non-zero only for finitely many i in I.
> Claim
> -----
> F^I is isomorphic (as a vector space over F) to finite(F^I) if and only
> if I is finite.
> Thoughts
> --------
> In addition to the proof, some interesting questions arise:
> 1) What is the cardinality of F^I?

|F|^|I| :-)

> 2) What could be a basis for F^I?
> 3) What is the dimension of F^I?

Let V be a vector space over the field F. If |V| is infinite and
|V| > |F| then the dimension of V over F is |V|.

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