I have a question about the following result from a paper that I am currently reading.
Let $X$ be a Hilbert space with inner product $\langle \cdot, \cdot\rangle$, $C$ be a finite dimensional subspace of $X$, and $S$ be a bounded subset of $X$. Suppose a nonlinear mapping $F:X\to X^*$ satisfies: $$\langle F(x)-F(y),z \rangle\le c|x-y||z|,\quad x,y,z\in S,$$ then the author claims that $$\langle PF(x)-PF(y),z \rangle\le c|x-y||z|\quad x,y,z\in S,$$ where $P:X\to C$ is the projection operator onto $C$.
I was wondering whether this is correct. I thought this follows from $$\langle Px,z \rangle \le \langle x,z \rangle ,x,z\in S$$ which is incorrect, since in that paper, there is no inclusion relation between $S$ and $C$.
Consider the dual operator of P.