Let $H$ be a Hilbert Space with the norm $||\,\cdot\,||$ and inner product $(\, ,\,)$. Define $I : H\to \mathbb{R}$ as a nonlinear functional on $H$.
Definition 1(Differentiability of I)
$I$ is differentiable at $u\in H$ if and only if $\exists v=I'[u]\in H$ such that $\forall w\in H, I[w] = I[u] + (v,w-u) + o(||w-u||)$.
Definition 2(Palais-Smale Condition)
$I \in C^{1}(H;\mathbb{R})$ satisfies Palais-Smale Compactness Condition (PS) if and only if $\forall \{u_{k}\}_{k=1}^{\infty}\subset H$ such that $\{I[u_{k}]\}_{k=1}^{\infty}$ is bounded and $I'[u_{k}]\to 0$ in $H$ as $k\to\infty$ is precompact in $H$.
Some important notations :
1. $\mathscr{C}:=\{I \,|\, I\in C^{1}(H;\mathbb{R}\}$
2. $A_{c} := \{u \in H\, | \, I[u] \leq c\}$
3. $K_{c} := \{u \in H\, | \, I[u]=c, I'[u]=0\}$
So, my question is how to show that $I'$ is bounded on a bounded set of $H$? I am uncertain how to show the claim. Here $I$ satisfies both Definition 1 and Definition 2 Any help is much appreciated, thank you!