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

Each result cited is universally quantified over the data in its own statement, and is applied here to the data named in the statement above; results stated for a Euclidean dimension $n$ are applied with $n=d$, the dimension fixed in Optimal Transport on the Flat Torus: Standing Notation §conventions.

Preliminaries. The measure $\mu$ belongs to $\mathcal{P}(\mathbb{R}^{d})$ by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures. Write $H=L^{2}(\mu;\mathbb{R}^{d})$. By Optimal Transport on the Flat Torus: Standing Notation §fields, $H$ is the space of classes of square-integrable random vectors in $\mathbb{R}^{d}$ on the probability space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$, and it is a real Hilbert space by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations §hilbert; by The Space of Square-Integrable Random Vectors §inner-product its inner product is $\langle\cdot,\cdot\rangle_{\mu}$ and $\lVert\cdot\rVert_{\mu}$ is the associated norm, so that $0\le\lVert\xi\rVert_{\mu}$ and $\lVert\xi\rVert_{\mu}^{2}=\langle\xi,\xi\rangle_{\mu}$ for $\xi\in H$. Write $d_{\mu}(\xi,\eta)=\lVert\xi-\eta\rVert_{\mu}$ for its distance, a metric by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric, and $0_{H}$ for its zero vector. Convergence of sequences in $H$, closed sets and the closure are those of Real Hilbert Space §topology, which is the sense in which Optimal Transport on the Flat Torus: Standing Notation §fields takes closures; by Sequential Characterization of the Closure in a Metric Space, a point of $H$ lies in the closure of a subset $A\subseteq H$ exactly when it is the limit of a sequence of points of $A$. A subset $A$ of $H$ carries the restriction of $d_{\mu}$, a metric on $A$ by claim 1 of The Restriction of a Metric to a Subset Induces the Subspace Topology; the resulting metric space is written $(A,d_{\mu})$. By The Space of Square-Integrable Random Vectors §classes, $[X]+[Y]=[X+Y]$ and $a[X]=[aX]$, hence $a[X]+b[Y]=[aX+bY]$, for square-integrable $X,Y$ and real $a,b$, and $0_{H}$ is the class of the constant map whose value is the origin; the additive inverse of $[X]$ is $[(-1)X]$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §vector-space. Consequently an identity between maps that holds at every point of $\mathbb{R}^{d}$ holds between their classes. As in the statement, $\nabla f$ also denotes the class in $H$ of the gradient of $f\in C^{1}_{\mathrm{per}}$, which lies in $H$ by Optimal Transport on the Flat Torus: Standing Notation §calculus. Finally, a sequence of real numbers has at most one limit: if $(a_{m})$ converges to $A$ and to $B$, then claim 1 of Order Properties of Limits of Real Sequences, applied with $b_{m}=a_{m}$ for every $m$, gives $A\le B$ and, with the roles of $A$ and $B$ exchanged, $B\le A$.

Step 1: claim 1. Let $f,g\in C^{\infty}_{\mathrm{per}}$ and let $a,b$ be real. By Elementary Properties of Lattice-Periodic Functions §algebra, which is in force by The Flat Torus: Standing Notation §periodic, the scalar multiples $af$ and $bg$ lie in $C^{\infty}_{\mathrm{per}}$, and therefore so does their sum $af+bg$. Every element of $C^{\infty}_{\mathrm{per}}$ is smooth on $\mathbb{R}^{d}$, that is, of class $C^{k}$ on $\mathbb{R}^{d}$ for every natural number $k$; being $\mathbb{Z}^{d}$-periodic, it therefore lies in $C^{1}_{\mathrm{per}}$ and in $C^{2}_{\mathrm{per}}$ by Lattice-Periodic Functions and the Periodic Function Classes §classes, so that its gradient and its Laplacian are defined as in Optimal Transport on the Flat Torus: Standing Notation §calculus. The set $\mathbb{R}^{d}$ is open in itself, as recorded in the preamble of Lattice-Periodic Functions and the Periodic Function Classes.

Fix $x\in\mathbb{R}^{d}$ and $i\in[d]$. Since $f$ and $g$ are of class $C^{1}$ on $\mathbb{R}^{d}$, their partial derivatives with respect to the $i$th variable exist at $x$, by clause 1 of C^k Maps on a Euclidean Open Set read through the scalar convention of its clause 3. Claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, with $U=\mathbb{R}^{d}$, applied first to the scalar multiples $af$ and $bg$ and then to their sum, shows that the partial derivative of $af+bg$ with respect to the $i$th variable exists at $x$ and that

$$\partial_{i}(af+bg)(x)=a\,\partial_{i}f(x)+b\,\partial_{i}g(x). \tag{1}$$

By The Flat Torus: Standing Notation §periodic and Gradient of a Real-Valued Function on a Euclidean Open Set, the gradient of a function of class $C^{1}$ at $x$ is the point of $\mathbb{R}^{d}$ whose $i$th component is its $i$th partial derivative at $x$; the $i$th component of $a\nabla f(x)+b\nabla g(x)$ is $a\,\partial_{i}f(x)+b\,\partial_{i}g(x)$, sums and scalar multiples of points of $\mathbb{R}^{d}$ being formed componentwise (sum of points, scalar multiple). By (1) and claim 1 of Euclidean Points as Tuples of Real Numbers, $\nabla(af+bg)(x)=a\nabla f(x)+b\nabla g(x)$; as $x$ was arbitrary, $\nabla(af+bg)=a\nabla f+b\nabla g$.

For the Laplacian, (1) holds at every $x$, so the function $\partial_{i}(af+bg)$ on $\mathbb{R}^{d}$ is the pointwise combination $a\,\partial_{i}f+b\,\partial_{i}g$. Since $f$ and $g$ are of class $C^{2}$ on $\mathbb{R}^{d}$, the functions $\partial_{i}f$ and $\partial_{i}g$ are of class $C^{1}$ on $\mathbb{R}^{d}$ by clause 2 of C^k Maps on a Euclidean Open Set, so their partial derivatives with respect to the $i$th variable exist at every point; claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, applied to $\partial_{i}f$ and $\partial_{i}g$ exactly as above, gives, with the iterated partial derivatives of clause 4 of C^k Maps on a Euclidean Open Set,

$$\partial_{i}\partial_{i}(af+bg)(x)=a\,\partial_{i}\partial_{i}f(x)+b\,\partial_{i}\partial_{i}g(x)\qquad(x\in\mathbb{R}^{d},\ i\in[d]).$$

Summing over $i\in[d]$ and using The Laplacian of a Twice Continuously Differentiable Function §laplacian together with claims 2 and 3 of Properties of Finite Sums, $\Delta(af+bg)(x)=a\,\Delta f(x)+b\,\Delta g(x)$ for every $x$, that is, $\Delta(af+bg)=a\Delta f+b\Delta g$.

Step 2: $G_{\mu}$ is a linear subspace. Let $z:\mathbb{R}^{d}\to\mathbb{R}$ be the zero function. It is $\mathbb{Z}^{d}$-periodic, trivially, and smooth on $\mathbb{R}^{d}$ by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set with the constant $0$, so $z\in C^{\infty}_{\mathrm{per}}$ by Lattice-Periodic Functions and the Periodic Function Classes §classes. Since $z=0z+0z$, Step 1 with $f=g=z$ and $a=b=0$ gives $\nabla z=0\nabla z+0\nabla z$; at each point every component of the right-hand side is $0\cdot\partial_{i}z(x)+0\cdot\partial_{i}z(x)=0$, so $\nabla z(x)$ is the origin for every $x$ (claim 1 of Euclidean Points as Tuples of Real Numbers), and the class of $\nabla z$ is $0_{H}$. Thus $0_{H}\in G_{\mu}$. If $g_{1},g_{2}\in G_{\mu}$ and $a,b$ are real, choose $f_{1},f_{2}\in C^{\infty}_{\mathrm{per}}$ with $g_{1}=\nabla f_{1}$ and $g_{2}=\nabla f_{2}$ (The Tangent Space of the Torus Wasserstein Space at a Probability Measure §gradients); by the Preliminaries and Step 1, $ag_{1}+bg_{2}$ is the class of $a\nabla f_{1}+b\nabla f_{2}=\nabla(af_{1}+bf_{2})$ with $af_{1}+bf_{2}\in C^{\infty}_{\mathrm{per}}$, so $ag_{1}+bg_{2}\in G_{\mu}$. Taking $a=b=1$, and then $b=0$, shows that $G_{\mu}$ is closed under sums and scalar multiples; so $G_{\mu}$ is a linear subspace of $H$.

Step 3: $T_{\mu}$ is a closed linear subspace, and $G_{\mu}$ is dense in it. By The Tangent Space of the Torus Wasserstein Space at a Probability Measure §tangent, $T_{\mu}$ is the closure of $G_{\mu}$ in $H$; by claims 1 and 2 of The Closure is the Smallest Closed Superset, applied in the topological space of Real Hilbert Space §topology, it contains $G_{\mu}$ and is closed. It contains $0_{H}$, which lies in $G_{\mu}$ by Step 2. Let $\xi,\eta\in T_{\mu}$ and let $a,b$ be real. By the Preliminaries there are sequences $(g_{m})_{m\in\mathbb{N}}$ and $(h_{m})_{m\in\mathbb{N}}$ in $G_{\mu}$ converging to $\xi$ and to $\eta$. By Step 2, $ag_{m}+bh_{m}\in G_{\mu}$ for every $m$, and by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §linear-limits (for the scalar multiples, and then for the sum) the sequence $(ag_{m}+bh_{m})$ converges to $a\xi+b\eta$, which therefore lies in $T_{\mu}$. With $a=b=1$ and with $b=0$ this shows that $T_{\mu}$ is closed under sums and scalar multiples, so it is a linear subspace that is closed: a closed linear subspace of $H$. This proves claim 2.

For later use, $G_{\mu}$ is dense in the metric space $(T_{\mu},d_{\mu})$: every $\xi\in T_{\mu}$ is the limit in $(H,d_{\mu})$ of a sequence in $G_{\mu}\subseteq T_{\mu}$, and that sequence also converges to $\xi$ in $(T_{\mu},d_{\mu})$, since convergence involves only the distances $d_{\mu}(g_{m},\xi)$, which are the same numbers in both spaces; by Sequential Characterization of the Closure in a Metric Space applied in $(T_{\mu},d_{\mu})$, $\xi$ lies in the closure of $G_{\mu}$ in $(T_{\mu},d_{\mu})$. That closure is a subset of $T_{\mu}$, so it equals $T_{\mu}$.

Step 4: claim 3. Let $\ell$ and $C$ be as in claim 3.

(a) A functional on $G_{\mu}$. For $g\in G_{\mu}$ choose $f\in C^{\infty}_{\mathrm{per}}$ with $g=\nabla f$ and put $\tilde\ell(g)=\ell(f)$. This does not depend on the choice: if also $g=\nabla f'$ with $f'\in C^{\infty}_{\mathrm{per}}$, then $f-f'=1f+(-1)f'$ lies in $C^{\infty}_{\mathrm{per}}$ and $\nabla(f-f')=\nabla f+(-1)\nabla f'$ by Step 1, whose class is $g+(-1)g=g-g=0_{H}$ by the Preliminaries. Hence $\lVert\nabla(f-f')\rVert_{\mu}=\lVert0_{H}\rVert_{\mu}=0$ by Elementary Identities in a Real Inner Product Space §zero, and the hypotheses on $\ell$ give $|\ell(f)-\ell(f')|=|\ell(1f+(-1)f')|\le C\cdot0=0$; since $0\le|\ell(f)-\ell(f')|$, antisymmetry of the order gives $|\ell(f)-\ell(f')|=0$, whence $\ell(f)=\ell(f')$ by claim 1 of Properties of the Absolute Value in an Ordered Field. If $g=\nabla f$ and $h=\nabla f'$ lie in $G_{\mu}$ and $a,b$ are real, then $ag+bh=\nabla(af+bf')$ with $af+bf'\in C^{\infty}_{\mathrm{per}}$ (Step 1 and the Preliminaries), so $\tilde\ell(ag+bh)=\ell(af+bf')=a\,\tilde\ell(g)+b\,\tilde\ell(h)$; and $|\tilde\ell(g)|=|\ell(f)|\le C\lVert\nabla f\rVert_{\mu}=C\lVert g\rVert_{\mu}$. Consequently, as $g-h=g+(-1)h\in G_{\mu}$,

$$|\tilde\ell(g)-\tilde\ell(h)|=|\tilde\ell(g+(-1)h)|\le C\,\lVert g-h\rVert_{\mu}=C\,d_{\mu}(g,h),$$

so $\tilde\ell$ is Lipschitz with constant $C$ from $(G_{\mu},d_{\mu})$ to $(\mathbb{R},d_{\mathbb{R}})$, hence uniformly continuous on $G_{\mu}$ by A Lipschitz Map is Uniformly Continuous.

(b) Extension to $T_{\mu}$. By Steps 2 and 3, $G_{\mu}$ is nonempty and dense in the metric space $(T_{\mu},d_{\mu})$, so Extension of a Uniformly Continuous Real Function from a Dense Subset §existence yields $L:T_{\mu}\to\mathbb{R}$, continuous on $T_{\mu}$, with $L(g)=\tilde\ell(g)$ for every $g\in G_{\mu}$. By Continuity Between Metric Spaces is Equivalent to Sequential Continuity §on-subset, applied with $A$ equal to the whole space $(T_{\mu},d_{\mu})$, the sequence $(L(y_{m}))$ converges to $L(y)$ whenever $(y_{m})$ is a sequence in $T_{\mu}$ converging to $y\in T_{\mu}$; by the argument of Step 3, a sequence in $T_{\mu}$ converging in $H$ converges in $(T_{\mu},d_{\mu})$ to the same limit. Let $\xi,\eta\in T_{\mu}$ and $a,b$ real, and choose sequences $(g_{m})$, $(h_{m})$ in $G_{\mu}$ converging to $\xi$ and $\eta$. Then $ag_{m}+bh_{m}\to a\xi+b\eta$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §linear-limits, so $(L(ag_{m}+bh_{m}))$ converges to $L(a\xi+b\eta)$; but $L(ag_{m}+bh_{m})=\tilde\ell(ag_{m}+bh_{m})=a\,L(g_{m})+b\,L(h_{m})$ by (a), and this sequence converges to $aL(\xi)+bL(\eta)$ by claims 3 and 1 of Arithmetic of Limits of Real Sequences. By uniqueness of limits (Preliminaries), $L(a\xi+b\eta)=aL(\xi)+bL(\eta)$. Moreover $|L(g_{m})|=|\tilde\ell(g_{m})|\le C\lVert g_{m}\rVert_{\mu}$ for every $m$ by (a); the left side converges to $|L(\xi)|$ by claim 4 of Order Properties of Limits of Real Sequences, and the right side converges to $C\lVert\xi\rVert_{\mu}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity and claim 3 of Arithmetic of Limits of Real Sequences; claim 1 of Order Properties of Limits of Real Sequences gives $|L(\xi)|\le C\lVert\xi\rVert_{\mu}$ for every $\xi\in T_{\mu}$. In particular $L(0_{H})=0$, by linearity with $a=b=0$.

(c) Existence by the Riesz representation. By Step 3, $T_{\mu}$ is a closed linear subspace of the real Hilbert space $H$, so, as recorded in the preamble of Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space, Nearest-Point Projection onto a Nonempty Closed Convex Subset of a Real Hilbert Space §existence provides the orthogonal projection $P=P_{T_{\mu}}:H\to H$, with $P\zeta\in T_{\mu}$ for every $\zeta\in H$. By Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §linear, $P$ is a linear map with $P\zeta=\zeta$ for $\zeta\in T_{\mu}$, and by Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §pythagoras, $\lVert P\zeta\rVert_{\mu}\le\lVert\zeta\rVert_{\mu}$. Define $\hat\ell:H\to\mathbb{R}$ by $\hat\ell(\zeta)=L(P\zeta)$. It is additive and homogeneous, as the composite of the linear maps $P$ and $L$ (the latter linear by (b)), and

$$|\hat\ell(\zeta)|\le C\,\lVert P\zeta\rVert_{\mu}\le C\,\lVert\zeta\rVert_{\mu}\qquad(\zeta\in H)$$

by (b) and by claim 5 of Elementary Arithmetic in an Ordered Field with the nonnegative $C$; so $\hat\ell$ is a bounded linear functional on $H$. By The Riesz Representation Theorem for a Real Hilbert Space §existence there is $\xi\in H$ with $\hat\ell(\zeta)=\langle\zeta,\xi\rangle_{\mu}$ for every $\zeta\in H$.

The point $\xi$ lies in $T_{\mu}$. Indeed, let $\zeta$ belong to the orthogonal complement $T_{\mu}^{\perp}$. Since $0_{H}\in T_{\mu}$ and $\zeta-0_{H}=\zeta\in T_{\mu}^{\perp}$, Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §characterisation gives $P\zeta=0_{H}$, so $\langle\zeta,\xi\rangle_{\mu}=\hat\ell(\zeta)=L(0_{H})=0$, hence $\langle\xi,\zeta\rangle_{\mu}=0$ by the symmetry of the inner product, condition (a) of Real Inner Product Space §inner-product. As $\zeta\in T_{\mu}^{\perp}$ was arbitrary, $\xi\in(T_{\mu}^{\perp})^{\perp}$, which equals $T_{\mu}$ by Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §complement.

For $f\in C^{\infty}_{\mathrm{per}}$, the class $\nabla f$ lies in $G_{\mu}\subseteq T_{\mu}$, so $P\nabla f=\nabla f$ and

$$\langle\xi,\nabla f\rangle_{\mu}=\langle\nabla f,\xi\rangle_{\mu}=\hat\ell(\nabla f)=L(\nabla f)=\tilde\ell(\nabla f)=\ell(f),$$

using symmetry of the inner product, the choice of $\xi$, the definition of $\hat\ell$, (b) and (a). Moreover, by claim 3 of Properties of the Absolute Value in an Ordered Field and the bound of this step,

$$\lVert\xi\rVert_{\mu}^{2}=\langle\xi,\xi\rangle_{\mu}=\hat\ell(\xi)\le|\hat\ell(\xi)|\le C\,\lVert\xi\rVert_{\mu}.$$

If $0<\lVert\xi\rVert_{\mu}$, then $\lVert\xi\rVert_{\mu}^{-1}$ exists and is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field, and multiplying by it, by claim 5 of Elementary Arithmetic in an Ordered Field, gives $\lVert\xi\rVert_{\mu}\le C$; otherwise $\lVert\xi\rVert_{\mu}=0\le C$, the norm being nonnegative. Thus $\xi\in T_{\mu}$ has the two asserted properties.

(d) Uniqueness. Let $\xi'\in T_{\mu}$ also satisfy $\langle\xi',\nabla f\rangle_{\mu}=\ell(f)$ for every $f\in C^{\infty}_{\mathrm{per}}$, and put $\zeta=\xi-\xi'$, which lies in $T_{\mu}$ by Step 3. For every $g\in G_{\mu}$, written $g=\nabla f$, one has $\langle\zeta,g\rangle_{\mu}=\langle\xi,\nabla f\rangle_{\mu}-\langle\xi',\nabla f\rangle_{\mu}=\ell(f)-\ell(f)=0$ by Elementary Identities in a Real Inner Product Space §bilinear. Choose a sequence $(g_{m})$ in $G_{\mu}$ converging to $\zeta$ (Preliminaries). By The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity, the sequence $(\langle g_{m},\zeta\rangle_{\mu})$ converges to $\langle\zeta,\zeta\rangle_{\mu}$; but $\langle g_{m},\zeta\rangle_{\mu}=\langle\zeta,g_{m}\rangle_{\mu}=0$ for every $m$, and the constant sequence with every term $0$ converges to $0$ in the sense of Limit of a Sequence of Real Numbers, since $|0-0|=0<\varepsilon$ for every positive $\varepsilon$ and every index. By uniqueness of limits, $\lVert\zeta\rVert_{\mu}^{2}=\langle\zeta,\zeta\rangle_{\mu}=0$; since $\lVert\zeta\rVert_{\mu}$ and $0$ are nonnegative with the same square, $\lVert\zeta\rVert_{\mu}=0$ by claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, so $\zeta=0_{H}$ by Elementary Identities in a Real Inner Product Space §vanishing, that is, $\xi'=\xi$. This proves claim 3.