I think I understand what both Hamel basis and Schauder basis mean. But to me, the Schauder basis makes more sense intuitively than the Hamel basis does. For example, Fourier series, Bases for Hilbert Spaces used in Quantum Mechanics – are all Schauder bases.
As I understand, even for simple infinite dimensional spaces ( e.g. $\ell^p$), a Hamel basis can have an uncountable cardinality and cannot be explicitly constructed. On the other hand, we can construct a Schauder basis for such spaces trivially.
So, my question is that despite this fact,
why is the concept of basis introduced as Hamel basis and not as Schauder basis?
(Specifically, is there some objective reason like even though Schauder basis seems superficially simpler, its mathematics gets complex going ahead? Or maybe some properties defined for finite-dimensional vector spaces don't carry forward?)
Best Answer
As soon as you have a vector space you can talk about a linear basis for it (i.e. a Hamel basis) in the usual sense (independent and spanning, in terms of finite linear combinations). So in a sense it's an elementary notion but it's not very useful for infinite-dimensional spaces (functions spaces, sequence spaces etc.) because they're too big and cannot be given explicitly in many cases.
But these vector spaces have a topology (metric/norm etc.) and we can talk about limits of finite linear combinations so Schauder bases become possible and can be given explicitly often. These often turn out to have extra applications (Fourier series etc.)
So if you're an algebraist or set theorist you study Hamel bases, in analysis and more application oriented directions people study Schauder bases.