Given functions $f,g: \mathcal{X} \to \mathcal{H}$ lying in a vector-valued RKHS $\mathcal{H}_v$, i.e. outputting functions in the RKHS $\mathcal{H}$, what can we do about bounding the quantity
$$\mathbb{E}_{x \sim \mathbb{P}_X}\left[ \langle f(x), g(x) \rangle_{\mathcal{H}} \right]$$
using Cauchy-Schwarz.
Do we have something like:
$$\mathbb{E}_{x \sim \mathbb{P}_X}\left[ \langle f(x), g(x) \rangle_{\mathcal{H}} \right] \le \sqrt{\mathbb{E}_{x \sim \mathbb{P}_X}\left[ \Vert f(x) \Vert_{\mathcal{H}}^2 \right]\mathbb{E}_{x \sim \mathbb{P}_X}\left[ \Vert g(x) \Vert_{\mathcal{H}}^2 \right]}$$
What can we say about
$$\mathbb{E}_{x \sim \mathbb{P}_X}\left[ h(x) \Vert f(x) - g(x) \Vert_{\mathcal{H}}^2 \right]$$
for some some function $h:\mathcal{X} \to \mathbb{R}$?
