Question regarding basis and dimension So I was reading Linear Algebra: A geometrical approach by S.Kumaresan and there  is a problem saying prove that a vector space with finite number of elements in basis will be finite dimensional. Though it is easy to prove, in the very next part they said although in later part you will see that converse is not necessarily true. And here is my problem. Considering what they are saying then a finite dimensional vector space may have infinite number of elements in basis. Consider a vector space $V$ which is finite dimensional(let it be $n$ dimensional) have an infinite set as basis. Now there is a definition that statesA vector space is $k$ dimensional if it has a set of $k$ elements as basis. so the vector space $V$ which is finite dimensional(let it be $n$ dimensional) will have a finite element($n$) set as basis. But it is not true. Since in a finite dimensional vector space's any two bases have same number of elements. hence all of vector space$V$'s basis will have n element which is contradictory by our first assumption. Then we can't say the vector space is finite dimensional. But that seems contradictory too. I mean a vector space can't be finite dimensional and infinite dimensional at the same time because according to definition of finite dimensional: A vector space$V$ is finite dimensional if it has a finite elements set $S$ such that $L(S)=V$. Where $L(S)$ is span of the set $S$. Please tell me where am I wrong in this whole arguement. Thanks in advance.
OK. So these are the pictures of the definitions and theorems I have used on question.
OK. So these are the pictures of the definitions and theorems I have used on question.