Is every closed subspace of the Schwartz space densely embedded into its dual space?

Question

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 every 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 $\mathcal{S}$.

Answer 1

No to both questions. Consider $V=\{ g\in\mathcal S: g(-x)=ig(x) \textrm{ for }x\ge 0\}$. This space is closed and $V\not= 0$ because we can start with a compactly supported function on $[0,\infty)$, with support at some distance from $x=0$ and then define $g$ on $(-\infty, 0)$ as required.

Then $T_f=0\in V'$ for every $f\in V$ because $fg$ is odd when $f,g\in V$.

Answer 2

This is more of a long(ish) comment than an actual answer... The question(s) asked above are not the same as the one posed in the title. In the former you are referring only to closed subspaces of $\mathcal{S}$, whereas in the latter you seem to ask whether any (abstract) nuclear Fréchet space $V$ (i.e. not necessarily topologically embedded into $\mathcal{S}$) has a positive answer to the two questions posed above.

This is important because a.) any closed subspace of $\mathcal{S}$ can be identified with a closed subspace of the nuclear Fréchet space $s$ of sequences of rapid decrease (use e.g. the sequence of Hermite coefficients of elements of $\mathcal{S}$), and b.) a nuclear Fréchet space $V$ is (topologically isomorphic to) a closed subspace of $s$ if and only if it has the so-called dominating norm (DN) property: there is a system $\|\cdot\|_k$, $k\in\mathbb{N}$ of seminorms defining the topology of $V$ and a $p\in\mathbb{N}$ such that for all $k\in\mathbb{N}$ there are $C_k>0$, $n_k\in\mathbb{N}$ such that $$\|x\|^2_k\leq C_k\|x\|_p\|x\|_{n_k}\ .$$ See e.g. Proposition 31.5, pp. 395 of the book by R. Meise and D. Vogt, Introduction to Functional Analysis (Oxford University Press, 1997) for a proof. Not all nuclear Fréchet spaces possess this property - for instance, $V=C^\infty(\Omega)$, where $\varnothing\neq\Omega\subset\mathbb{R}^n$ is open, does not (see e.g. Corollary 31.13, pp. 401-402 of R. Meise and D. Vogt, ibid.). In view of that, it would perhaps be better to clarify either the question or at least its title, for the answer may depend on that.

Edit: with the updated title, the answer to both questions is generally no as pointed by Nick S's comment to Christian Remling's answer. I will leave it to Nick whether he wants to turn his comment into an actual answer.