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Math Forum » Discussions » sci.math.* » sci.math.independent

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

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Kaba

Posts: 289
Registered: 5/23/11
F^I isomorphic to finite(F^I)
Posted: Mar 5, 2013 6:03 PM
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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?

2) What could be a basis for F^I?

3) What is the dimension of F^I?

--
http://kaba.hilvi.org

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