I'm working on Reproducing Kernel Hilbert Spaces and I had a problem proving the the continuity of the evaluation functional $e_{t}$ ($e_{t}(\phi) = \phi(t)$).
Theorem
A Hilbert Space of complex valued functions on E has a reproducing kernel then all the evaluation functionals $e_{t}$, $t \in E$, are continuous on $\mathcal{H}$.
Proof: If $\mathcal{H}$ has a reproducing kernel $K$ then for any $t \in E$, we have
$$\forall \phi \in \mathcal{H} \ \ \ \ \ \ \ \ e_{t}(\phi) = \langle \phi, K(\cdot, t) \rangle_{\mathcal{H}}$$
Thus the evaluation functional $e_{t}$ is linear and by the Cauchy-Schwarz inequality.
$$|e_{t}(\phi)| = |\langle\phi, K(\cdot, t) \rangle_{\mathcal{H}}| \ \ \leq \ \ ||\phi|| \ \ ||K(\cdot, t)|| = ||\phi|| \ \ [K(t, t)]^{1/2}$$
$$ |e_{t}(\phi)| \leq ||\phi|| \ \ [K(t, t)]^{1/2}$$
Then $e_{t}$ is bounded then $e_{t}$ is continuous.
How can I conclude that $e_{t}$ is bounded from the last inequality? As far as I know $[K(t, t)]^{1/2}$ is not bounded.
Best Answer
You have the assumption that
$$\forall \phi \in \mathcal{H} \quad e_{t}(\phi) = \langle \phi, K(\cdot, t)\rangle _{\mathcal{H}}$$ Consider the meaning of the above statement. The thing on the right implicitly says that $K(\cdot,t)$ is an element of $\mathcal H$, because otherwise the inner product is not defined. Any functional of this form is bounded. Specifically, $$|e_{t}(\phi)| \le \|\phi\|_{\mathcal H}\, \| K(\cdot, t)\|_{\mathcal{H}}$$