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?