Consider the left regular representation of $\mathbb{R}$ in $\mathrm{L}^2(\mathbb{R})$. Let us denote by $\mathrm{L}^2(\mathbb{R})^\infty$ the algebraic subspace of smooth vectors, or equivalently, the Gårding subspace (the theorem of Dixmier-Malliavin assumed to be at our disposal). I want to prove

$$\mathrm{L}^2(\mathbb{R})^\infty = \{f\in\mathcal{C}^\infty(\mathbb{R}) \mid f^{(n)}\in\mathrm{L}^2(\mathbb{R})\:\:\text{for all}\:\:n\geq 0\}.$$

Actually, I proved the inclusion $\subseteq$ by making use of an exercise in Knapp's Reprensentation Theory of Semisimple Groups (Problem 5 on p. 57). My questions are: Is the above equality really true? If so, could you please give a reference, or an argument for the proof of $\supseteq$?

Consider the left regular representation of $\mathbb{R}$ in $\mathrm{L}^2(\mathbb{R})$. Let us denote by $\mathrm{L}^2(\mathbb{R})^\infty$ the algebraic subspace of smooth vectors, or equivalently, the Gårding subspace (the theorem of Dixmier-Malliavin assumed to be at our disposal). I want to prove

$$\mathrm{L}^2(\mathbb{R})^\infty = \{f\in\mathcal{C}^\infty(\mathbb{R}) \mid f^{(n)}\in\mathrm{L}^2(\mathbb{R})\:\:\text{for all}\:\:n\geq 0\}.$$

Actually, I proved the inclusion $\subseteq$ by making use of an exercise in Knapp's Representation Theory of Semisimple Groups (Problem 5 on p. 57). My questions are: Is the above equality really true? If so, could you please give a reference, or an argument for the proof of $\supseteq$?