The Torus Tangent Space: Linearity of the Periodic Calculus, Closed Subspace, and Representation of Bounded Functionals on Gradients

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let $\mu\in\mathcal{P}(\mathbb{T}^{d})$, with $L^{2}(\mu;\mathbb{R}^{d})$, $\langle\cdot,\cdot\rangle_{\mu}$ and $\lVert\cdot\rVert_{\mu}$ as in Optimal Transport on the Flat Torus: Standing Notation §fields, gradients and Laplacians as in Optimal Transport on the Flat Torus: Standing Notation §calculus, and $G_{\mu}$ and $T_{\mu}$ as in The Tangent Space of the Torus Wasserstein Space at a Probability Measure §gradients and The Tangent Space of the Torus Wasserstein Space at a Probability Measure §tangent. Then the following hold.

1. (Linearity of the periodic calculus)¶ For $f,g\in C^{\infty}_{\mathrm{per}}$ and real $a,b$, the function $af+bg$ belongs to $C^{\infty}_{\mathrm{per}}$, $\nabla(af+bg)=a\nabla f+b\nabla g$ and $\Delta(af+bg)=a\Delta f+b\Delta g$.

2. (Subspaces)¶ $G_{\mu}$ is a linear subspace of $L^{2}(\mu;\mathbb{R}^{d})$, and $T_{\mu}$ is a closed linear subspace of $L^{2}(\mu;\mathbb{R}^{d})$.

3. (Representation)¶ Let $\ell:C^{\infty}_{\mathrm{per}}\to\mathbb{R}$ satisfy $\ell(af+bg)=a\,\ell(f)+b\,\ell(g)$ for all $f,g\in C^{\infty}_{\mathrm{per}}$ and real $a,b$, and let $C\ge0$ be a real number with $|\ell(f)|\le C\lVert\nabla f\rVert_{\mu}$ for every $f\in C^{\infty}_{\mathrm{per}}$. Then there is exactly one $\xi\in T_{\mu}$ with $\langle\xi,\nabla f\rangle_{\mu}=\ell(f)$ for every $f\in C^{\infty}_{\mathrm{per}}$, and it satisfies $\lVert\xi\rVert_{\mu}\le C$.