While trying to distinguish the definitions of Hamel and Schauder bases of infinite dimensional vector spaces, I was told that the true definition of a Hamel basis is not "a spanning set with linearly independent vectors", but instead "a spanning set that guarantees unique representation of each vector in the space" (although both definitions coincide in the finite dimensional case).
The second definition seems stronger, so I seek an example, in the context of infinite dimensional vector spaces, of a Hamel basis by the first definition, but not by the second.
You say the two definitions are equivalent in the finite-dimensional case. How do you prove that? If you don't know how to prove it you're just believing what you're told - if you do know how to prove it you should note that finite dimension doesn't come up...