Doubt in the Hamel Basis of infinite Dimension Vector space

Question

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..

Answer 1

Yes, that is immediate from the definition of a Hamel basis. Every element is a finite linear combination of elments in the Hamel basis.

If a vector $x$ is not a finite linear combination of elements of the Hamel basis $H$ then we can see that $H \cup \{x\}$ would be linearly independent, contradicting the fact that $H$ is a maximal linearly independent set.

Answer 2

Yes, by the definition of Hamel basis. But note that, in this context, each expression of a vector $v$ as a linear combination of elements of the basis only involves finite sums: $v$ can be written as $a_1v_1+\cdots+a_nv_n$, where each $v_1,\ldots,v_n$ is an element of the basis.