# Doubt in the Hamel Basis of infinite Dimension Vector space

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..

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.
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.
• @gune They are using $\sum_i$ and $\sum_j$ denote finite sums. They could have used a more explicit notation but what they have written is also OK. Commented Oct 25, 2019 at 0:05