Proof of On the Real Line the Tangent Space is the Whole Space of Square-Integrable Vector Fields

Throughout, each result cited is universally quantified over the data appearing in its own statement and is applied to the data named here. By Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii) the triple $(\mathbb{R},\mathcal{B}(\mathbb{R}),\mu)$ is a probability space and $L^{2}(\mu;\mathbb{R})$ is the space of classes of square-integrable random variables on it, with $\langle\xi,\eta\rangle_{\mu}=\int_{\mathbb{R}}\xi\eta\,d\mu$ and $\lVert\xi\rVert_{\mu}^{2}=\int_{\mathbb{R}}\xi^{2}\,d\mu$, by claim 1 of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative. For $\psi\in C_{c}^{\infty}(\mathbb{R})$ claim 2 of that lemma gives $\nabla\psi=\psi'$ and $\langle\xi,\nabla\psi\rangle_{\mu}=\int_{\mathbb{R}}\xi\,\psi'\,d\mu$. Write $\lambda_{1}$ for the Lebesgue measure on $\mathcal{B}(\mathbb{R})$.

Since $T_{\mu}$ is a subset of $L^{2}(\mu;\mathbb{R})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients (i), it suffices to prove the reverse inclusion. Let $\eta\in L^{2}(\mu;\mathbb{R})$, and fix a Borel representative of it, again written $\eta$.

Step 1 (A tangent field with the same pairings). Define $\ell:C_{c}^{\infty}(\mathbb{R})\to\mathbb{R}$ by $\ell(\psi)=\langle\eta,\nabla\psi\rangle_{\mu}$. It is linear: for $\psi,\phi\in C_{c}^{\infty}(\mathbb{R})$ and $a,b\in\mathbb{R}$, The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure (iii) gives $\nabla(a\psi+b\phi)=a\nabla\psi+b\nabla\phi$, and claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion gives the linearity of the integral. By claim 1 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm, applied on the probability space $(\mathbb{R},\mathcal{B}(\mathbb{R}),\mu)$ to the square-integrable random variables $\eta$ and $\nabla\psi$,

$$|\ell(\psi)|\le\lVert\eta\rVert_{\mu}\,\lVert\nabla\psi\rVert_{\mu}\qquad\text{for every }\psi\in C_{c}^{\infty}(\mathbb{R}).$$

Hence Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients (iv), applied with $C=\lVert\eta\rVert_{\mu}$, provides $\xi\in T_{\mu}$ with $\langle\xi,\nabla\psi\rangle_{\mu}=\ell(\psi)$ for every $\psi\in C_{c}^{\infty}(\mathbb{R})$.

Put $w=\eta-\xi\in L^{2}(\mu;\mathbb{R})$, with a Borel representative again written $w$. By linearity of the inner product,

$$\int_{\mathbb{R}}w\,\psi'\,d\mu=0\qquad\text{for every }\psi\in C_{c}^{\infty}(\mathbb{R}). \tag{$*$}$$

If $\lVert w\rVert_{\mu}=0$ then $w$ is the zero class and $\eta=\xi\in T_{\mu}$, as required; so assume $0<\lVert w\rVert_{\mu}$.

Note that $w$ is integrable with respect to $\mu$, with $\int_{\mathbb{R}}|w|\,d\mu\le\lVert w\rVert_{\mu}$: this is claim 1 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm applied to $|w|$ and the constant random variable $1$, whose mean-square norm is $1$ since $\mu$ is a probability measure.

Step 2 (Pairing with mean-zero test functions). Let $\varphi\in C_{c}^{\infty}(\mathbb{R})$ with $\int_{\mathbb{R}}\varphi\,d\lambda_{1}=0$. By Test Functions on the Real Line: Unit Mass, Large Mass, Mean-Zero Correction, and the Primitive of a Mean-Zero Test Function (v) there is $\psi\in C_{c}^{\infty}(\mathbb{R})$ with $\psi'=\varphi$, so $(*)$ gives

$$\int_{\mathbb{R}}w\,\varphi\,d\mu=0 .$$

Step 3 (Pairing with an arbitrary test function). Let $\theta$ be as in Test Functions on the Real Line: Unit Mass, Large Mass, Mean-Zero Correction, and the Primitive of a Mean-Zero Test Function (ii), so $\theta\in C_{c}^{\infty}(\mathbb{R})$ is nonnegative with $\int_{\mathbb{R}}\theta\,d\lambda_{1}=1$, and put $a=\int_{\mathbb{R}}w\,\theta\,d\mu$, a real number because $\theta$ is bounded by A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable and $w$ is integrable.

Let $\psi\in C_{c}^{\infty}(\mathbb{R})$ and put $c=\int_{\mathbb{R}}\psi\,d\lambda_{1}$, a real number by Test Functions on the Real Line: Unit Mass, Large Mass, Mean-Zero Correction, and the Primitive of a Mean-Zero Test Function (i). By Test Functions on the Real Line: Unit Mass, Large Mass, Mean-Zero Correction, and the Primitive of a Mean-Zero Test Function (iv) the function $\varphi=\psi-c\,\theta$ is a test function with $\int\varphi\,d\lambda_{1}=0$, so Step 2 and claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion give

$$0=\int_{\mathbb{R}}w\,\varphi\,d\mu=\int_{\mathbb{R}}w\,\psi\,d\mu-c\,a,\qquad\text{that is}\qquad \int_{\mathbb{R}}w\,\psi\,d\mu=a\int_{\mathbb{R}}\psi\,d\lambda_{1}. \tag{$**$}$$

Step 4 (The constant vanishes). Let $\varepsilon$ be a positive real number and put $C=\lVert w\rVert_{\mu}\,\varepsilon^{-1}$, a positive real number by claim 7 of Elementary Order Arithmetic in an Ordered Field and claim 5 of that lemma. By Test Functions on the Real Line: Unit Mass, Large Mass, Mean-Zero Correction, and the Primitive of a Mean-Zero Test Function (iii) there is $\chi\in C_{c}^{\infty}(\mathbb{R})$ with $0\le\chi\le1$ and $C\le\int_{\mathbb{R}}\chi\,d\lambda_{1}$. Applying $(**)$ to $\chi$, and using $|w\chi|\le|w|$ pointwise together with claims 1 and 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion,

$$|a|\,C\le|a|\int_{\mathbb{R}}\chi\,d\lambda_{1}=\Bigl|\int_{\mathbb{R}}w\,\chi\,d\mu\Bigr|\le\int_{\mathbb{R}}|w|\,d\mu\le\lVert w\rVert_{\mu},$$

the first inequality by claim 5 of Elementary Arithmetic in an Ordered Field with the nonnegative factor $|a|$ and claim 4 of Properties of the Absolute Value in an Ordered Field. Multiplying by the positive $C^{-1}$ gives $|a|\le\lVert w\rVert_{\mu}C^{-1}=\varepsilon$. As $\varepsilon$ was an arbitrary positive real number, Comparison of Real Numbers with Arbitrary Positive Slack (iii) gives $|a|=0$, hence $a=0$ by claim 1 of Properties of the Absolute Value in an Ordered Field.

Step 5 (Conclusion). By $(**)$ with $a=0$,

$$\int_{\mathbb{R}}w\,\psi\,d\mu=0\qquad\text{for every }\psi\in C_{c}^{\infty}(\mathbb{R}).$$

Since $w$ is Borel and integrable with respect to $\mu\in\mathcal{P}(\mathbb{R})$, Test Functions Approximate Bounded Continuous Functions Pointwise, Determine a Finite Borel Measure, and Detect a Vanishing Density (iii), applied in dimension $d=1$ with $g=w$, gives $\mu(\{x\in\mathbb{R}:w(x)\ne0\})=0$. Hence $w$ is the zero class of $L^{2}(\mu;\mathbb{R})$ by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii), contradicting $0<\lVert w\rVert_{\mu}$; so that case does not occur, and $\eta=\xi\in T_{\mu}$.

Therefore $L^{2}(\mu;\mathbb{R})\subseteq T_{\mu}$, and with the reverse inclusion noted at the outset, $T_{\mu}=L^{2}(\mu;\mathbb{R})$.