Denote by \(\mathcal{H}^{1}\) the Cameron-Martin space (13.2) and by \(\mathcal{H}_{\circ}^{1}\) the set defined in (13.3).

a) Show that \(\mathcal{H}^{1}\) is a Hilbert space with the canonical norm \(\|h\|_{\mathcal{H}^{1}}^{2}:=\int_{0}^{1}\left|h^{\prime}(s)\right|^{2} d s\) and scalar product \(\langle\phi, hangle_{\mathcal{H}^{1}}=\int_{0}^{1} \phi^{\prime}(s) d h(s)\) (this is a Paley-WienerZygmund integral, cf. 13.5).

b) Show that \(\mathcal{H}_{\circ}^{1}\) is a dense subset of \(\mathcal{H}^{1}\).

c) Show that \(\|h\|_{\mathcal{H}^{1}}=\sup _{\phi \in \mathcal{H}^{1},\|\phi\|_{\mathcal{H}^{1}=1}}\langle\phi, hangle_{\mathcal{H}^{1}}=\sup _{\phi \in \mathcal{H}_{0}^{1},\|\phi\|_{\mathcal{H}^{1}}=1}\langle\phi, hangle_{\mathcal{H}^{1}}\).

d) Let \(h \in \mathcal{C}_{(0)}[0,1]\). Show that \(h otin \mathcal{H}^{1}\) if, and only if, there exists a sequence \(\left(\phi_{n}\right)_{n \geqslant 1} \subset \mathcal{H}_{\circ}^{1}\) such that \(\left\|\phi_{n}\right\|_{\mathcal{H}^{1}}=1\) and \(\left\langle\phi_{n}, h\rightangle_{\mathcal{H}^{1}} \geqslant 2 n\).

Data From  (13.2)

Data From  (13.3)

Transcribed Image Text:

1 := {u e e [ \u' (t) dt < o}. (13.2) Hue Co[0, 1] : u is a.c. and lu' (t) dt