My original question is from this ME post but I think I need a broader understanding for this.

The Schwartz space $\mathcal{S}$ and its subspaces are examples of nuclear spaces. In fact, any closed subspace of the Schwartz space is a nuclear Fréchet space.

Also, for any closed subspace $V$ of $\mathcal{S}$, we know that an element of $V$ may be regarded as an element of $V'$, the space of continuous linear functionals on $V$ by the mapping \begin{equation} f \to T_f \text{ where } T_f(g):= \int fg \text{ for all } g \in V. \end{equation}

Here are my questions now:

Is the above mapping always injective?

Is $V$ sequentially dense in $V'$ w.r.t. the weak$^*$ topology via the above mapping? That is, for each $T \in V'$, can we find a sequence $\{ f_n \} \subset V$ such that $\int f_n g \to T(g)$ as $n \to \infty$ for all $g \in V$?

Add : Following advice below, I restrict my question to closed subspaces of $\mathcal{S}$.