I don't think you can really call it a generalized dot product, because it doesn't map to a scalar. The inner product is the well accepted definition of a generalized dot product, and convolution does not follow the axioms that an inner product must follow.
Its nearly the same thing, isn't it? If you denote by Tx the left-shift operator defined by (Tx f)(y) = f(y+x), then the correlation of f and g evaluated at x is precisely the dot product of f and Tx g. If you evaluate your function at a certain point, you obtain a scalar product.
> (...) then the correlation of f and g evaluated at x
It really isn't the same, and oddly enough you unknowingly show that off, by mentioning that convolution is the function that maps input functions to the output function, but the dot product is at best a single point evaluated with the output function.
