I'm sorry if I sound too ignorant, i don't have a high level of knowledge in math
I've been lately trying to understand the Frèchet derivative. I'm just starting Calc II, but I have a tiny grasp in multivariable calculus. That is, I understand that one can treat some of the variables as constants and get a directional derivative.
According to wikipedia, if the limit of the equation below as $h$ tends to $0$ is equal to $0$ then the function is said to be Fréchet differentiable at $x$.
And as much as this can work as a definition, I'm still wondering what the Fréchet derivative actually is.
My confusion might also come from the fact that there are some ideas in multi-variable calculus I don't get, so I will try to summarize my questions below.
Q1. If I understood correctly, $f$ could be a multi-variable function, so how come we use only one $x$? Is it because $x$ is itself a multi-variable vector?
Q2. What is the meaning of $T(h)$? As in, what is it, and what is it doing in this equation?
Q3. Why does this equation -when $h$ tends to $0$- somehow tells whether a function is (Fréchet) differentiable or not? How does this equation relates to what a Fréchet derivative is?
Any help/thoughts would be really appreciated.