The space of polynomials over an infinite field (where the vectors are the coefficients of each power of x).

(each coordinate is either zero or one, and the addition is 0+0=1+1=0, 0+1=1+0=1). Over the field .

The space of continuous functions

spaces

The list goes on...

What sort of examples are you looking for. Try looking up Infinite Dimensional Vector Space examples.