Functions as "vectors" I'm reading a book called "Fourier series and orthogonal functions" by Davis since it seemed pretty readable (at first) and since I don't really know what is going on with these fourier series, yet. The book suggests that one can think of functions as vectors. After all a sequence is a function whose domain is the set of real numbers. And a vectors is a list of coordinates (x, y, z) or as many dimensions as you like. This "functions can be vectors" idea seems pretty central to the ideas in the book, and I'm getting confused about how ... for example ... a continuous function could be a vector. A continuous function is defined at every singe point and there's nothing discrete about the domain. Is the idea that through use of series we create a list-like correspondence that allows us to think of the function as a vector? I'm really confused. What kind of dimensional space would contain continuous functions as vectors? Wouldn't it need to have infinite dimensions?