Calculate the normal cone to $X=\{f: [0,1] \rightarrow [0,1]\mid f \text{ increasing}\} $

Let $X:=\{f: [0,1] \rightarrow [0,1]\mid f \text{ increasing}\} $. We endow it with $L_2$ norm, and thus $X \subset L_2([0,1])$.We can show that the set $X$ is closed and thus compact, and also convex.

We define the normal cone to $X$ at $f \in X$ as $N(f;X):=\{h^*\in L_2([0,1]) \mid \langle h^*,g-f\rangle \leq 0, \forall g\in X\}$.

For this normal cone $N(f;X)$, I have the following observations:

1. $N(f;X)=\{0^{[0,1]}\} $ for $f\in \mathbb{int}X$.

2. By taking $g=0^{[0,1]}$, we must have $\int_{0}^1 f(x)h^*(x) \mathbb{d}x\geq0 $

3. By taking $g=1^{[0,1]}$, we must have $\int_{0}^1 (1-f(x))h^*(x) \mathbb{d}x\leq0 $

My question is: how to calculate the $N(f;X)$ in the closed-form expression?

I cannot find any reference for this type of question.What I can only find is the papers discussing the normal cone of $Y:=\{f: [0,1] \rightarrow \mathbb{R} \mid f \text{ increasing}\} $ (e.g., Theorem 3.9 in Natile and Savaré (2009), see https://epubs.siam.org/doi/10.1137/090750809. ). But in these papers, $Y$ is a convex cone, which is quite different from what I want to deal with.

Thank you very much!