Cardinality of infinite dimensional vector space of functions

Question

Assume that F is an infinite field, k is an infinite cardinal, and V= {f: k→F| f is a function} 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?