What exactly is the difference between Gateaux derivative and directional derivative?

Question

The definition of the limit looks very similar between the two derivatives.

It seems that directional derivative is the "amount" of the function going in the direction of a vector (arrow), whereas the Gateaux derivative is the "amount" of a function going in the direction of another function - can someone verify if I have this correctly? Is there anything beyond this?

In addition can someone explain why it is necessary to state that the Gateaux derivative is defined on Banach spaces? Let's say I have a set of function that does not form a Banach space, intuitively why would that cause difficulty to define a derivative?

And when do I use Gateaux derivative when I am told to calculate the derivative of a function?

Answer 1

As I understand it, on infinite dimensional Banach spaces, there is a Gâteaux derivative and a Fréchet derivative. Both involve the idea $$ f(x+h) = f(x) + A(h) + r(h) ,$$ where $A$ is the derivative of $f$ at $x$. The difference is whether $r(th)/t \to 0$ as $t \to 0$ for each $h$ in the Banach space, or whether $r(h)/\|h\| \to 0$ as $h \to 0$.

You generally do this in a Banach space, because both involve limits. If your space isn't complete, then limits are less likely to exist. Since it is easy to take the completion, why not? (Well you might want $A$ to be an unbounded operator, in which case incomplete linear spaces may be more appropriate.)

On finite dimensional linear spaces, all these definitions end up being equivalent.

The wikipedia articles describe them well: https://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative, https://en.wikipedia.org/wiki/G%C3%A2teaux_derivative. Notice that the Gâteaux might not be linear.