Linearity of the Noise Gradient, and the Noise Tangent Space is a Closed Linear Subspace

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mu\in\mathcal{P}(X)$. The set $\mathcal{F}C^{1}_{b}(X)$ of bounded $C^{1}$ cylindrical functions, the noise gradient $\nabla_{a}\varphi$ of $\varphi\in\mathcal{F}C^{1}_{b}(X)$, the set $G^{a}_{\mu}$ of The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradients and the noise tangent space $T^{a}_{\mu}$ are those of The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space, and $L^{2}(\mu;X^{a})$ is the real Hilbert space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields. For $\varphi,\psi\in\mathcal{F}C^{1}_{b}(X)$ and $s,t\in\mathbb{R}$ the function $s\varphi+t\psi$ belongs to $\mathcal{F}C^{1}_{b}(X)$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §linear, so that its noise gradient is defined; maps into $X^{a}$ are added and multiplied by reals pointwise. Then the following hold.

1. (Linearity of the noise gradient)¶ For all $\varphi,\psi\in\mathcal{F}C^{1}_{b}(X)$ and $s,t\in\mathbb{R}$,

$$\nabla_{a}(s\varphi+t\psi)=s\,\nabla_{a}\varphi+t\,\nabla_{a}\psi .$$

2. (Linear subspaces)¶ $G^{a}_{\mu}$ is a linear subspace of $L^{2}(\mu;X^{a})$, and $T^{a}_{\mu}$ is a closed linear subspace of $L^{2}(\mu;X^{a})$.