Strange that this didn’t get the right answer back then. The right answer is that there are two notions of basis that are relevant here: Hamel basis, where every vector must be represented by a finite linear combination of basis vectors, and Schauder basis, where countably infinite linear combinations are allowed (which must converge as infinite sums, which means this notion doesn’t work on all vector spaces, only those with a topology or a norm).
Hamel basis for infinite-dimensional spaces is a pretty awkward notion (try writing out a Hamel basis of R as a vector space over Q). Schauder bases are often more natural and nicer to work with. For example, the Fourier series of a periodic function is its representation in a countable basis of sines and cosines.
A typical infinite-dimensional Hilbert space is L^2, the space of functions that are measurable and whose square is integrable. (Or rather equivalence classes of functions, because two functions that differ on a set of measure 0 represent the same element of L^2.) It’s not quite the set of all continuous functions—there can be a bunch of discontinuities—but it’s still much “smaller” than the set of all functions, and admits a countable basis in the Schauder sense. (In fact if we take functions on a circle, the Fourier basis works.) And you can also prove that it has no countable basis in the Hamel sense.
Strange that this didn’t get the right answer back then. The right answer is that there are two notions of basis that are relevant here: Hamel basis, where every vector must be represented by a finite linear combination of basis vectors, and Schauder basis, where countably infinite linear combinations are allowed (which must converge as infinite sums, which means this notion doesn’t work on all vector spaces, only those with a topology or a norm).
Hamel basis for infinite-dimensional spaces is a pretty awkward notion (try writing out a Hamel basis of R as a vector space over Q). Schauder bases are often more natural and nicer to work with. For example, the Fourier series of a periodic function is its representation in a countable basis of sines and cosines.
A typical infinite-dimensional Hilbert space is L^2, the space of functions that are measurable and whose square is integrable. (Or rather equivalence classes of functions, because two functions that differ on a set of measure 0 represent the same element of L^2.) It’s not quite the set of all continuous functions—there can be a bunch of discontinuities—but it’s still much “smaller” than the set of all functions, and admits a countable basis in the Schauder sense. (In fact if we take functions on a circle, the Fourier basis works.) And you can also prove that it has no countable basis in the Hamel sense.