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**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 $\displaystyle [0,2\pi],$ 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.