Originally Posted by Ackbeet
In infinite-dimensional vector spaces, there can be functions in the space which require an infinite sum of vectors in the basis. Just think about periodic functions on the interval $[0,2\pi],$ with the Fourier sines and cosines as a basis set. If I take a function as simple as the triangle function, I'm going to need an infinite sum of sines and cosines in order to write the triangle function as a linear combination of sines and cosines. If you consider the continuous functions on the interval in question, this is a vector space.

This example doesn' apply: the OP talked about vector spacew whereas your example is a vector space with an inner product. The first one is an abstract algebraic object and nothing more, and the second one is an abstract algebraic object TOGETHER with an inner product, a norm, a distance , etc.
In abstract vector spaces ONLY there's no meaninful way to define an infinite sum of vectors.

Tonio