I don't know if it's true but I think in a finite dimensional veccter space, an orthonormal set which is complete becomes a basis. But in a Hilbert space, the books say that the set of finite linear combinations of the elements in a complete orthonormal set is only dense in the Hilbert space. So why aren't they equal? Does the problem lie in closedness or infinity of dimension? Would you please give me an example of a Hilbert space where a complete orthonormal set is not a basis?
Thanks to everyone.