Math Help - Intuitive understanding of Frechet derivative?

So I know that a directional derivative df(a)/dv can be thought of as the rate of change of the function at the point a in the direction of the unit vector v.

But what is the Frechet derivative supposed to be? How is it related to a directional derivative. I have seen it described as a linear mapping of f at a point a. It seems to be able to map functions into different dimensions, such as R2 -> R1, R1 -> R3, etc...As it is called Frechet derivative I take it is a measure of the rate of change of something...but rate of change of what? How am I supposed to visualize the Frechet derivative?

Any help is appreciated.

Anybody able to explain the Frechet derivative? (In plain english if possible)