> a continuous wave function is equivalent to a countably infinite vector space I think you mean that the function is a point in the vector space. More precise

> a continuous wave function is equivalent to a countably infinite vector space I think you mean that the function is a point in the vector space. More precisely, it is a point in a particular separable Hilbert space, i.e., a complete vector space equipped with an inner product that has countable, dense basis.