# nonlinearity after projection operator

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 $C$.

• How does $PF(x)$ make sense? If $F:X \to X^*$ and $P:X \to C$ then $F(x) \in X^*$, and so what does $P$ do to an element of $X^*$? (Assuming that $PF(x) = P(F(x))$) Aug 13, 2017 at 17:50
• I think that it is by identifying $X^*$ as $X$ Aug 13, 2017 at 17:52
• @FlybyNight I think the author identified $X$ as $X^*$.
– John
Aug 13, 2017 at 17:55

• I see. And use the fact $\|P^*\|=\|P\|$.