I'm currently reading $\textit{An Introduction do the Theory of Reproducing Kernel Hilbert Spaces}$ by $\textit{Vern I. Paulsen}$.
Let $f\in \mathcal{H}(K)$ with $||f||$=c. The following inequality is used frequently: $$f(x)\overline{f(y)}\leq c^{2}K(x,y)$$ Why does this inequality hold?
Using the Cauchy-Schwarz-Inequality I get $$ |f(x)\overline{f(y)}|=|\langle f,k_{x}\rangle \langle k_{y},f \rangle| \leq \langle f,f\rangle \langle k_{x},k_{x} \rangle \langle k_{y},k_{y} \rangle\langle f,f\rangle =c^{2}\langle k_{x},k_{x} \rangle \langle k_{y},k_{y}\rangle$$ which is not exactly what I what. Any suggestions?
This is a typo. What Paulsen means is $(f(x)\overline{f(y)})\leq (K(x,y))$, or in other words, $(x,y)\mapsto K(x,y)-f(x)\overline{f(y)}$ is a kernel function. That this is true is content of Theorem 4.15 (at least in his notes online).