Vector spases of functions are infinite-dimensional. For example, the space $C[0,1]$ consisting of all continuous functions on the segment [0,1], $C^k[0,1]$ consiting of all k-times differentialble functions on [0,1], Lebesgue spases $L_p(\Omega)$, Sobolev spaces $W^m_p$, and many other spaces are infinite-dimensional. This spaces arise in the theory of differential equations.

Fourier series, for example.

Fourier Series and Transforms