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[Community Question] Linear-algebra: Cardinality of infinite dimensional vector space of functions

One of our user asked:

Assume that F is an infinite field, k is an infinite cardinal, and V= F^k is a vector space. Can it be prove proved that |V|= dimV?
My thought was that we know that|V|= max {dimV,|F|}, so all I need is to prove is that if |V|=|F|, then also |F|=dimV. Now, from Konig theorem, if |V|=|F|, then k< cf(|F|), so I tried to prove that k< cf(|F|) is imposible, but i didn't know how to continue this line of thougt. Has sombodey know how to prove it?


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