For all $(h_1, h_2) \in \mathcal{H} \times \mathcal{H}$, we denote $(h_1 \mid h_2)_{\mathcal{H}} = c_\lambda \left(D(h_1) \mid D(h_2)\right)$ where $c_\lambda$ is introduced in question (11a). (a) Verify that $(\mid)_{\mathcal{H}}$ defines an inner product on $\mathcal{H}$. (b) Show that for all $x \in \mathbf{R}$ and $h \in \mathcal{H}$ we have $h(x) = \left(\tau_x(\gamma_{2\lambda}) \mid h\right)_{\mathcal{H}}$. (c) Show that for all $h \in \mathcal{H}$ we have $$\|h\|_\infty \leq \|h\|_{\mathcal{H}}$$ where we have set $\|h\|_\infty = \sup_{x \in \mathbf{R}} |h(x)|$ and $\|h\|_{\mathcal{H}} = (h \mid h)_{\mathcal{H}}^{1/2}$.
For all $(h_1, h_2) \in \mathcal{H} \times \mathcal{H}$, we denote $(h_1 \mid h_2)_{\mathcal{H}} = c_\lambda \left(D(h_1) \mid D(h_2)\right)$ where $c_\lambda$ is introduced in question (11a).
(a) Verify that $(\mid)_{\mathcal{H}}$ defines an inner product on $\mathcal{H}$.
(b) Show that for all $x \in \mathbf{R}$ and $h \in \mathcal{H}$ we have $h(x) = \left(\tau_x(\gamma_{2\lambda}) \mid h\right)_{\mathcal{H}}$.
(c) Show that for all $h \in \mathcal{H}$ we have
$$\|h\|_\infty \leq \|h\|_{\mathcal{H}}$$
where we have set $\|h\|_\infty = \sup_{x \in \mathbf{R}} |h(x)|$ and $\|h\|_{\mathcal{H}} = (h \mid h)_{\mathcal{H}}^{1/2}$.