I know that by Zorn's Lemma we can prove that every vector space has a Hamel Basis. Where Hamel Basis means a maximal Linearly independent set.
My question is, if this is the Finite Dimension case, then we will be able to write any element of the vector space as a linear combination of the Basis vectors.
But is it also true for the infinite Dimension case?
I came across this doubt when reading the last line of this attached picture (Funtional Analysis By J.B. Conway):
So for me it looks like they have written any vector of $X$ as a sum of the Hamel Basis Elements..