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## Main Question or Discussion Point

In analogy to vector spaces, can we define a set of "basis functions" from which any continuous function can be written as a (possibly infinite) linear combination of the basis functions?

I know the trigonometric functions 1, sin(nx), cos(nx) can be used for monotonic continuous functions, but not every Fourier series leads to a convergent solution (sinx+sin2x+sin3x+... for example diverges).

Is there a set of functions (not necessarily orthogonal) that spans all continuous functions and does not contain divergent series?

I know the trigonometric functions 1, sin(nx), cos(nx) can be used for monotonic continuous functions, but not every Fourier series leads to a convergent solution (sinx+sin2x+sin3x+... for example diverges).

Is there a set of functions (not necessarily orthogonal) that spans all continuous functions and does not contain divergent series?