In the "Lectures on the Ekeland Variational Principle with Applications and Detours", the theorem 5.1 have that :
Let $X$ is Banach space, $\Phi:X \rightarrow \mathbb{R}$ is $C^1$. $K$ is a metric compact space. $K_0\subset K$ closed and $K_0\neq K$. Let $f_0:K_0 \rightarrow X$. Assume:$$\Gamma=\{f\in C(K,X)|f=f_0 \textit{ in } K_0\}\neq \varnothing$$
And $$\max_{K_0}\Phi<\max_K\Phi$$
We have : $$\min_{\mu\in\partial \Theta(x)}\sup_{k\in C(K,X),||k||_{\infty}\leq 1}\left<\mu,\left<\Phi^{'}(f_{\varepsilon}(.)),k(.)\right>\right>\leq \varepsilon.$$ So that : $$\min_{\mu\in\partial \Theta(x)}\left<\mu,\sup_{k\in C(K,X),||k||_{\infty}\leq 1}\left<\Phi^{'}(f_{\varepsilon}(.)),k(.)\right>\right>\leq \varepsilon$$.
My question is why ?
There are a lot of theorem to come to my question so please help me !