Alright, so I'm reading a book on Hilbert spaces and functional analysis, and here it defines a "Hamel basis" to be a "maximal linearly independent set". I take this to mean that $S$ is a Hamel basis if $S$ is linearly independent, and there is no linearly independent set $T$ for which $|T|>|S|$.
On the other hand, this article defines (on page 6) a "Hamel basis" to be a basis in which "we do not allow infinite sums" (i.e. in which every element can be expressed as a finite linear combination of basis vectors).
Also, the distinction they make between an orthonormal basis and a Hamel basis makes me think that an orthonormal basis does allow infinite sums, but their definition of "basis" on the first page requires finite linear combinations as well.
I'm just really confused, I can't tell which source I should trust and can't seem to find a single resource that gives the "correct" answer (not sure if I want to trust Wikipedia, though they seem to make the distinction too when defining the Schauder basis). What's the correct definition of a Hamel basis? Does the general definition of a "basis" allow for infinite linear combinations, or only finite ones? What about orthonormal and Hamel? Why would the book I'm reading define a Hamel basis in the way it did? What's even the point of making the distinction between a Hamel basis and any other basis; is a Hamel basis also orthonormal?
To completely avoid ambiguity, these are the rigorous definitions I'm using:
Linearly Independent: A subset $S$ of a vector space $V$ over a field $K$ is linearly independent if for every finite subset $G\subseteq S$, the only sequence $\{\alpha_s\}_{s\in G}$ to $\sum_{s\in G}\alpha_ss = 0$ is $\{\alpha_s\} = 0$.
Basis: A subset $S$ of a vector space $V$ is a basis if it is linearly independent, and every vector $v\in V$ can be expressed as a finite linear combination of vectors in $S$.
Orthonormal Basis: A subset $S$ of a Hilbert space $\Bbb{H}$ is an orthonormal basis if it is pairwise orthogonal ($e_1,\ e_2\in S$ and $e_1\neq e_2\rightarrow e_1\perp e_2$), every element is a unit vector, and every element of $\Bbb{H}$ can be expressed as a linear combination (finite or infinite) of elements of $S$.
Hamel Basis Definition 1: A Hamel basis for a Hilbert space $\Bbb{H}$ is a set $S$ such that $S$ is linearly independent, and there is no linearly independent set $T$ with $|T|>|S|$.
Hamel Basis Definition 2: A Hamel basis for a Hilbert space is a (potentially orthonormal?) basis under finite linear combinations.
