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Question about basis in infinite dimensional vector spaces?

I read that in infinite dimensional vector spaces, a countable ortonormal system is considered a basis if the set of finite linear combiantions of elements of such system is everywhere dense in the vector space. For example, the set {ei / i in N} is a basis for l^2 (oo) (where ei is the sequence with a 1 in the i-th location and 0 everywhere else). I was wondering if there was a way of considering a set a basis if every element in the space is a finite linear combination of the elements of the set and this set is linearly independent. I guess the vector space itself generates the vector space, but it's elements are not linearly independent. Is there a way to remove some of the elements of the vector space in such a way that the set that remains is linearly independent and it generates all the space only with finite combinations?
gpt
These two notions of "Basis" are different. Technically, a *Basis* is a set of linearly independent vector where every other vector in the vector space can be written as a *finite* linear combination of elements from the basis. In this case, your "basis" for l^(2) is not an actual basis, since we would need infinite combinations of these things. The Axiom of Choice guarantees that there is an actual Basis for every vector space, but it's not always possible to explicitly find them.

If we're in a more geometric setting, we can look at a different kind of basis called a "Continuous Basis", which is what you describe. In this way, we can write every vector as a convergent infinite linear combination of basis vectors. You need the extra geometry to talk about converging sequences like this. Generally, these are the kinds of bases that you find when doing Fourier Analysis or in Functional Analysis in general. So, while l^(2) does have a basis, it's not very helpful, but it does have a continuous basis that helps us understand the space as an Inner Product Space rather than just as a Vector Space.