Spaces of the derivative in a direction

Question

I have two question regarding the spaces where the first, and second, directional derivatives of a functional are.

Let $\Omega \in \mathbb{R}^2$ an open subset. Let the functional:

$$\phi =L^p (\Omega) \rightarrow \mathbb{R}$$ such that $u \mapsto \phi (u)$. If I get its derivative in the $v \in H_0 ^1 (\Omega)$ direction at the point $u$ it is: $$\phi ' (u) \cdot v$$ and it belongs to a space that I will call $X$.

Then, if I get the second derivative of $\phi$, based on its first, it is, at a given direction $w \in H_0 ^1 (\Omega)$:

$$\left \langle \phi '' (u) \cdot v , w \right \rangle _{X , H_0 ^1 (\Omega)}$$

So my questions are:

1. What space is the one that I named $X$?
2. To which space the second derivative $\left \langle \phi '' (u) \cdot v , w \right \rangle _{X , H_0 ^1 (\Omega)}$ belongs?

Thank you for your answers.