Unlike the reals considered as a vector space over the rationals, I know of a number of nice examples of vector spaces with uncountable dimension that have a nice basis.

For example, the space of rational functions on the real or complex numbers with complex coefficients whose numerator degree does not exceeds the denominator degree has as basis the fractional functions $1/(1+sx)^k$ with arbitrary complex $s$ and positive integers $k$.

Another example is the space of linear combinations of exponential functions $e^{sx}$ on the real or complex numbers.

Is there a common generalization? Or even some sort of general theory behind these examples?

Edit: Given the discussion so far, let me make the question more precise: Is there a fairly general way of proving that certain nice uncountable families of nice functions on some nice domain have the property that any finite subset is linearly independent? One kind of niceness criterion (but not the only relevant one) could be that the space spanned by the functions from the family can be characterizd in a basis-free way. Another niceness criterion is that the proof of linear independence should be not completely trivial.

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