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

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, in dimension $d=1$, with the identification of $\mathbb{R}$ and $\mathbb{R}^{1}$ and the notation of claims 1 and 2 of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative in force. Let $\mu\in\mathcal{P}_{2}(\mathbb{R})$, let $L^{2}(\mu;\mathbb{R})$ be the space of square-integrable vector fields against $\mu$ and let $T_{\mu}$ be the tangent space at $\mu$.

1. (The tangent space is everything)(i) $T_{\mu}=L^{2}(\mu;\mathbb{R})$.