Proof of Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients

Each result cited is universally quantified over the data in its own statement. Write $H=L^{2}(\mu;\mathbb{R}^{d})$, a real Hilbert space by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, with inner product $\langle\cdot,\cdot\rangle_{\mu}$, norm $\lVert\cdot\rVert_{\mu}$ and distance $d_{\mu}(\xi,\eta)=\lVert\xi-\eta\rVert_{\mu}$ (Real Inner Product Space §distance); the notation of Real Hilbert Spaces: Standing Notation and Background is in force for $H$, in particular convergence of sequences in $H$ is convergence in $(H,d_{\mu})$, closure is that of Real Hilbert Space §topology, and The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity, Nearest-Point Projection onto a Nonempty Closed Convex Subset of a Real Hilbert Space, Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space and The Riesz Representation Theorem for a Real Hilbert Space are in force by Real Hilbert Spaces: Standing Notation and Background §background. By Sequential Characterization of the Closure in a Metric Space, a point of $H$ lies in $\overline{A}$ exactly when it is the limit of a sequence in $A$. Operations on classes are formed on representatives (The Space of Square-Integrable Random Vectors §classes), so an identity between maps that holds pointwise on $\mathbb{R}^{d}$ holds between their classes. The natural number $n$ is read as a real number through the canonical map of The Canonical Map from the Natural Numbers to a Field, which is strictly increasing by claim 6 and takes positive values by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, so that $\psi_{R}$ is defined for $R=n$; each $\psi_{n}$ is a test function, as recorded in the statement.

Step 1: claim 1. The zero map is a test function whose gradient is the zero map, by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §linear with $a=b=0$; its class is the zero vector of $H$. For $\psi,\phi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and $a,b\in\mathbb{R}$, the same claim gives $a\nabla\psi+b\nabla\phi=\nabla(a\psi+b\phi)$ with $a\psi+b\phi$ a test function, so $G_{\mu}$ contains the zero vector and is closed under sums and scalar multiples: it is a linear subspace of $H$. By claims 1 and 2 of The Closure is the Smallest Closed Superset, $T_{\mu}=\overline{G_{\mu}}$ contains $G_{\mu}$ and is closed. It is a linear subspace: it contains the zero vector, and if $\xi,\eta\in T_{\mu}$ and $a,b\in\mathbb{R}$, choose sequences $(g_{m})$, $(h_{m})$ in $G_{\mu}$ converging to $\xi$ and $\eta$; then $ag_{m}+bh_{m}\in G_{\mu}$ and $ag_{m}+bh_{m}\to a\xi+b\eta$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §linear-limits, so $a\xi+b\eta\in T_{\mu}$. Hence $T_{\mu}$ is a closed linear subspace. Finally, $(T_{\mu},d_{\mu})$ is a metric space by Real Hilbert Spaces: Standing Notation and Background §topology, and every $\xi\in T_{\mu}$ is the limit in $(H,d_{\mu})$ of a sequence in $G_{\mu}$, which is then also its limit in $(T_{\mu},d_{\mu})$ (the distances are the same numbers); 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 that space, so this closure is $T_{\mu}$, that is, $G_{\mu}$ is dense in $(T_{\mu},d_{\mu})$.

Step 2: claim 2. The identity map is continuous from $(\mathbb{R}^{d},d_{E})$ to itself (take $\delta=\varepsilon$ in Continuous Map Between Metric Spaces), hence Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces. The function $x\mapsto\lVert\mathrm{id}(x)\rVert^{2}=\lVert x\rVert^{2}$ is the nonnegative Borel function of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs, whose integral is $M_{2}(\mu)<\infty$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment and The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space; as its negative part is zero, it is integrable with integral $M_{2}(\mu)$ by Integrable Function and the Lebesgue Integral. Thus $\mathrm{id}$ is a square-integrable random vector on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$ in the sense of The Space of Square-Integrable Random Vectors §space, and its class lies in $H$.

Let $M$ be the constant of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §test for the chosen $\chi$. For $n\in\mathbb{N}$ and $x\in\mathbb{R}^{d}$ put $f_{n}(x)=\lVert\nabla\psi_{n}(x)-x\rVert^{2}$. Since $\nabla\psi_{n}$ and $\mathrm{id}$ are square-integrable random vectors on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$, so is their pointwise difference $\nabla\psi_{n}-\mathrm{id}$ (The Space of Square-Integrable Random Vectors §space), which is therefore Borel, and $f_{n}=\lVert\nabla\psi_{n}-\mathrm{id}\rVert^{2}$ is the random variable of Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §composition, hence Borel; also $0\le f_{n}(x)$ for every $x$ by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and claim 2 of Nonnegativity of Squares in an Ordered Field, so $|f_{n}(x)|=f_{n}(x)$. For every $i\in[d]$, $|\partial_{i}\psi_{n}(x)-x_{i}|\le|\partial_{i}\psi_{n}(x)|+|x_{i}|\le M\lVert x\rVert+\lVert x\rVert$ by claim 5 of Properties of the Absolute Value in an Ordered Field, Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §test and claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, so by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers,

the last equality by claim 3 of Properties of Finite Sums and The Canonical Map from the Natural Numbers to a Field; $g$ is integrable, being a scalar multiple of the integrable function $\lVert x\rVert^{2}$ (claim 2 of Linearity and Monotonicity of the Lebesgue Integral). Fix $x$. By claim 1 of The Archimedean Property of the Real Numbers there is $n_{0}\in\mathbb{N}$ with $\lVert x\rVert<n_{0}$, and for $n\ge n_{0}$ we have $n_{0}\le n$ as real numbers, so $\lVert x\rVert<n$ (claim 2 of Elementary Order Arithmetic in an Ordered Field) and, by Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §ball, $\partial_{i}\psi_{n}(x)=x_{i}$ for every $i$, whence $f_{n}(x)=0$. Thus $f_{n}(x)\to0$ for every $x$, and claim 3 of Dominated Convergence Theorem gives $\int_{\mathbb{R}^{d}}f_{n}\,d\mu\to\int_{\mathbb{R}^{d}}0\,d\mu=0$ (the integral of the zero function vanishing by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space), that is, $\lVert\nabla\psi_{n}-\mathrm{id}\rVert_{\mu}^{2}\to0$, the integral of the nonnegative function $f_{n}$ in the sense of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures agreeing with its integral as an integrable function by claim 6(c) of Borel Measurability and Bounded Integration on a Metric Space. Given $\varepsilon>0$, eventually $\lVert\nabla\psi_{n}-\mathrm{id}\rVert_{\mu}^{2}<\varepsilon^{2}$, hence $\lVert\nabla\psi_{n}-\mathrm{id}\rVert_{\mu}<\varepsilon$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; so $\lVert\nabla\psi_{n}-\mathrm{id}\rVert_{\mu}\to0$, which says that $\nabla\psi_{n}\to\mathrm{id}$ in $(H,d_{\mu})$. Since each $\nabla\psi_{n}\in G_{\mu}$, Sequential Characterization of the Closure in a Metric Space gives $\mathrm{id}\in\overline{G_{\mu}}=T_{\mu}$.

Step 3: claim 3. By Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §test, $|\Delta\psi_{n}(x)|\le dM$ for all $n$ and $x$, and the constant $dM$ is integrable with respect to $\mu$ with $\int dM\,d\mu=dM\,\mu(\mathbb{R}^{d})=dM$ by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space and $\mu(\mathbb{R}^{d})=1$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures); each $\Delta\psi_{n}$ is Borel by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §laplacian. For fixed $x$ and $n\ge n_{0}$ as in Step 2, $\Delta\psi_{n}(x)=d$ by Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §ball, so $\Delta\psi_{n}(x)\to d$. By claim 3 of Dominated Convergence Theorem, $\int\Delta\psi_{n}\,d\mu\to\int d\,d\mu=d\,\mu(\mathbb{R}^{d})=d$, the constant integral again by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space.

Step 4: claim 4. (a) The functional on $G_{\mu}$. For $g\in G_{\mu}$ choose $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ with $g=\nabla\psi$ and put $\tilde\ell(g)=\ell(\psi)$. This does not depend on the choice: if also $g=\nabla\phi$, then $\nabla(\psi-\phi)=\nabla\psi-\nabla\phi$ (The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §linear) is the zero vector of $H$, whose norm is $0$ by Elementary Identities in a Real Inner Product Space §zero, so $|\ell(\psi)-\ell(\phi)|=|\ell(\psi-\phi)|\le C\lVert\nabla(\psi-\phi)\rVert_{\mu}=0$ and $\ell(\psi)=\ell(\phi)$ by claim 1 of Properties of the Absolute Value in an Ordered Field. For $g=\nabla\psi$, $h=\nabla\phi$ in $G_{\mu}$ and $a,b\in\mathbb{R}$, $ag+bh=\nabla(a\psi+b\phi)$ gives $\tilde\ell(ag+bh)=a\tilde\ell(g)+b\tilde\ell(h)$, and $|\tilde\ell(g)|\le C\lVert g\rVert_{\mu}$. Consequently $|\tilde\ell(g)-\tilde\ell(h)|=|\tilde\ell(g-h)|\le C\,d_{\mu}(g,h)$, 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 Step 1, $G_{\mu}$ 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 $g\in G_{\mu}$. Continuity on $(T_{\mu},d_{\mu})$ is sequential continuity by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §on-subset. Let $\xi,\eta\in T_{\mu}$, $a,b\in\mathbb{R}$, and choose sequences $(g_{m})$, $(h_{m})$ in $G_{\mu}$ with $g_{m}\to\xi$, $h_{m}\to\eta$ in $H$, hence in $(T_{\mu},d_{\mu})$. Then $ag_{m}+bh_{m}\to a\xi+b\eta$ (The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §linear-limits), so $L(a\xi+b\eta)=\lim_{m}L(ag_{m}+bh_{m})=\lim_{m}\bigl(aL(g_{m})+bL(h_{m})\bigr)=aL(\xi)+bL(\eta)$ by claims 3 and 1 of Arithmetic of Limits of Real Sequences; and $|L(\xi)|=\lim_{m}|L(g_{m})|\le\lim_{m}C\lVert g_{m}\rVert_{\mu}=C\lVert\xi\rVert_{\mu}$, using claim 4 of Order Properties of Limits of Real Sequences for $|L(g_{m})|\to|L(\xi)|$, The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity for $\lVert g_{m}\rVert_{\mu}\to\lVert\xi\rVert_{\mu}$, claim 3 of Arithmetic of Limits of Real Sequences for $C\lVert g_{m}\rVert_{\mu}\to C\lVert\xi\rVert_{\mu}$, and claim 1 of Order Properties of Limits of Real Sequences.

(c) Riesz representation. Let $P=P_{T_{\mu}}$ be the orthogonal projection onto the closed linear subspace $T_{\mu}$ (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu), which satisfies $P\zeta\in T_{\mu}$ for every $\zeta\in H$ (Nearest-Point Projection onto a Nonempty Closed Convex Subset of a Real Hilbert Space §existence), is linear with $P\zeta=\zeta$ for $\zeta\in T_{\mu}$ (Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §linear), and satisfies $\lVert P\zeta\rVert_{\mu}\le\lVert\zeta\rVert_{\mu}$ (Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §pythagoras). Define $\hat\ell:H\to\mathbb{R}$ by $\hat\ell(\zeta)=L(P\zeta)$. It is linear, as a composition of linear maps, and $|\hat\ell(\zeta)|\le C\lVert P\zeta\rVert_{\mu}\le C\lVert\zeta\rVert_{\mu}$, 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 $\zeta\in H$. If $\zeta\in T_{\mu}^{\perp}$ (Orthogonality, Orthogonal Complement and Orthonormal Families in a Real Inner Product Space §complement), then $P\zeta$ is the zero vector by Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §characterisation (the zero vector lies in $T_{\mu}$ and $\zeta-0\in T_{\mu}^{\perp}$), so $\langle\zeta,\xi\rangle_{\mu}=L(0)=0$ ($L$ is linear), hence $\langle\xi,\zeta\rangle_{\mu}=0$ by the symmetry of the inner product (Real Inner Product Space §inner-product); as $\zeta\in T_{\mu}^{\perp}$ was arbitrary, $\xi\in(T_{\mu}^{\perp})^{\perp}=T_{\mu}$ by Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §complement. For $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$, since $\nabla\psi\in T_{\mu}$, $\langle\xi,\nabla\psi\rangle_{\mu}=\langle\nabla\psi,\xi\rangle_{\mu}=\hat\ell(\nabla\psi)=L(\nabla\psi)=\tilde\ell(\nabla\psi)=\ell(\psi)$, the symmetry of the inner product being part of Real Inner Product Space §inner-product. Moreover $\lVert\xi\rVert_{\mu}^{2}=\langle\xi,\xi\rangle_{\mu}=\hat\ell(\xi)\le C\lVert\xi\rVert_{\mu}$ (Real Inner Product Space §norm); if $\lVert\xi\rVert_{\mu}>0$, multiplying by its inverse (positive by claim 7 of Elementary Order Arithmetic in an Ordered Field; claim 5 of Elementary Arithmetic in an Ordered Field) gives $\lVert\xi\rVert_{\mu}\le C$, and otherwise $\lVert\xi\rVert_{\mu}=0\le C$.

(d) Uniqueness. Let $\xi'\in T_{\mu}$ also satisfy $\langle\xi',\nabla\psi\rangle_{\mu}=\ell(\psi)$ for every $\psi$, and put $\zeta=\xi-\xi'\in T_{\mu}$ (Step 1). Then $\langle\zeta,g\rangle_{\mu}=0$ for every $g\in G_{\mu}$ by Elementary Identities in a Real Inner Product Space §bilinear. Choose a sequence $(g_{m})$ in $G_{\mu}$ with $g_{m}\to\zeta$; then $\lVert\zeta\rVert_{\mu}^{2}=\langle\zeta,\zeta\rangle_{\mu}=\lim_{m}\langle\zeta,g_{m}\rangle_{\mu}=0$ by Real Inner Product Space §norm and The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity, so $\lVert\zeta\rVert_{\mu}=0$ and $\zeta$ is the zero vector by Elementary Identities in a Real Inner Product Space §vanishing, that is, $\xi'=\xi$.

Step 5: claim 5. By Elementary Identities in a Real Inner Product Space §bilinear, $\langle\eta-\xi,\nabla\psi\rangle_{\mu}=\ell(\psi)-\ell(\psi)=0$ for every $\psi$, so $\eta-\xi$ is orthogonal to every element of $G_{\mu}$, and, by the limit argument of Step 4(d), to every element of $T_{\mu}$: $\eta-\xi\in T_{\mu}^{\perp}$. Since $\xi\in T_{\mu}$, Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §characterisation gives $\xi=P_{T_{\mu}}\eta$, and Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §pythagoras gives $\lVert\xi\rVert_{\mu}\le\lVert\eta\rVert_{\mu}$. If $\eta\in T_{\mu}$, then $P_{T_{\mu}}\eta=\eta$ by Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §linear, so $\xi=\eta$.