[proofplan] We use the standard Otto-Villani implication from a logarithmic Sobolev inequality to a quadratic transport-[entropy inequality](/theorems/6729). The standard Gaussian measure satisfies the logarithmic Sobolev inequality with constant $1$. Substituting this constant into the [Otto-Villani theorem](/theorems/6794) gives exactly $W_2^2(\nu,\gamma_n)\leq 2H(\nu\mid\gamma_n)$. We then check the extended-real cases and note that finite entropy automatically gives the finite second moment needed for the quadratic Wasserstein distance to be finite. [/proofplan] [step:Record the two standard inputs with constants] For a nonnegative measurable function $u$ on $\mathbb{R}^n$ with $\int_{\mathbb{R}^n}u\,d\gamma_n<+\infty$, define its entropy with respect to $\gamma_n$ by \begin{align*} \operatorname{Ent}_{\gamma_n}(u) := \int_{\mathbb{R}^n}u\log u\,d\gamma_n - \left(\int_{\mathbb{R}^n}u\,d\gamma_n\right) \log\left(\int_{\mathbb{R}^n}u\,d\gamma_n\right), \end{align*} with the convention $0\log 0=0$. We use the following standard form of the Gaussian logarithmic Sobolev inequality: for every locally Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$ such that the displayed quantities are finite, \begin{align*} \operatorname{Ent}_{\gamma_n}(f^2) \leq 2\int_{\mathbb{R}^n}|\nabla f|^2\,d\gamma_n. \end{align*} Thus $\gamma_n$ satisfies the logarithmic Sobolev inequality with constant $C=1$ in the convention \begin{align*} \operatorname{Ent}_{\mu}(f^2)\leq 2C\int_{\mathbb{R}^n}|\nabla f|^2\,d\mu. \end{align*} We also use the standard Otto-Villani theorem in this same normalization: if a probability measure $\mu$ on $\mathbb{R}^n$ satisfies the logarithmic Sobolev inequality with constant $C>0$, then for every Borel probability measure $\eta$ on $\mathbb{R}^n$, \begin{align*} W_2^2(\eta,\mu)\leq 2C\,H(\eta\mid\mu). \end{align*} The conclusion uses the extended-real convention for both sides. [guided] For a nonnegative measurable function $u$ on $\mathbb{R}^n$ with \begin{align*} \int_{\mathbb{R}^n}u\,d\gamma_n<+\infty, \end{align*} define \begin{align*} \operatorname{Ent}_{\gamma_n}(u) := \int_{\mathbb{R}^n}u\log u\,d\gamma_n - \left(\int_{\mathbb{R}^n}u\,d\gamma_n\right) \log\left(\int_{\mathbb{R}^n}u\,d\gamma_n\right), \end{align*} with the convention $0\log 0=0$. The Gaussian logarithmic Sobolev inequality says that every locally Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$ for which the displayed terms are finite satisfies \begin{align*} \operatorname{Ent}_{\gamma_n}(f^2) \leq 2\int_{\mathbb{R}^n}|\nabla f|^2\,d\gamma_n. \end{align*} Thus $\gamma_n$ has logarithmic Sobolev constant $C=1$ in the normalization \begin{align*} \operatorname{Ent}_{\mu}(f^2)\leq 2C\int_{\mathbb{R}^n}|\nabla f|^2\,d\mu. \end{align*} In the same normalization, the Otto-Villani theorem says that if a probability measure $\mu$ on $\mathbb{R}^n$ satisfies the logarithmic Sobolev inequality with constant $C>0$, then every Borel probability measure $\eta$ on $\mathbb{R}^n$ satisfies \begin{align*} W_2^2(\eta,\mu)\leq 2C\,H(\eta\mid\mu). \end{align*} The conclusion uses the extended-real convention. Therefore the constants match exactly: the Gaussian has $C=1$, and Otto-Villani will produce the coefficient $2C=2$. [/guided] [/step] [step:Apply Otto-Villani to the Gaussian measure] Let $\nu$ be a Borel probability measure on $\mathbb{R}^n$. Applying the Otto-Villani theorem from the previous step with $\mu=\gamma_n$, $\eta=\nu$, and $C=1$ gives \begin{align*} W_2^2(\nu,\gamma_n)\leq 2H(\nu\mid\gamma_n). \end{align*} This is the desired estimate. It remains only to make explicit why no hidden finite-moment hypothesis is being used. If $H(\nu\mid\gamma_n)=+\infty$, then the inequality is immediate in the extended-real sense. Suppose instead that $H(\nu\mid\gamma_n)<+\infty$. Then $\nu\ll\gamma_n$. Let $h=d\nu/d\gamma_n$, and fix $a\in(0,1/2)$. The entropy variational inequality applied to the measurable function $x\mapsto a|x|^2$ gives \begin{align*} \int_{\mathbb{R}^n}a|x|^2\,d\nu(x) \leq H(\nu\mid\gamma_n) + \log\int_{\mathbb{R}^n}\exp(a|x|^2)\,d\gamma_n(x). \end{align*} The Gaussian exponential moment in the last display is finite for $a\in(0,1/2)$. Hence \begin{align*} \int_{\mathbb{R}^n}|x|^2\,d\nu(x)<+\infty. \end{align*} Since $\gamma_n$ also has finite second moment, $W_2(\nu,\gamma_n)$ is finite whenever the entropy is finite. Thus the displayed Otto-Villani estimate covers all Borel probability measures $\nu$. [/step]