Noise Gradients of Bounded $C^2$ Cylindrical Functions Are Dense in the Noise Tangent Space

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mathcal{F}C^{1}_{b}(X)$ be the set of bounded $C^{1}$ cylindrical functions and $\mathcal{F}C^{2}_{b}(X)$ the set of bounded $C^{2}$ cylindrical functions. For $\varphi\in\mathcal{F}C^{1}_{b}(X)$, $\nabla_{a}\varphi$ is its noise gradient. For $\mu\in\mathcal{P}(X)$, $L^{2}(\mu;X^{a})$ is the real Hilbert space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, with norm $\lVert\cdot\rVert_{\mu}$, the class of $\nabla_{a}\varphi$ in it is again written $\nabla_{a}\varphi$, and $T^{a}_{\mu}$ is the noise tangent space at $\mu$.

1. (Inclusion)¶ $\mathcal{F}C^{2}_{b}(X)\subseteq\mathcal{F}C^{1}_{b}(X)$. In particular, for every $\psi\in\mathcal{F}C^{2}_{b}(X)$ and every $\mu\in\mathcal{P}(X)$ the noise gradient $\nabla_{a}\psi$ is defined and its class belongs to $T^{a}_{\mu}$.

2. (Density)¶ Let $\mu\in\mathcal{P}(X)$ and $v\in T^{a}_{\mu}$. There is a sequence $(\psi_{j})_{j\in\mathbb{N}}$ in $\mathcal{F}C^{2}_{b}(X)$ with

$$\lim_{j\to\infty}\lVert\nabla_{a}\psi_{j}-v\rVert_{\mu}=0 .$$