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?
> 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|.