I have a question regarding separability of a certain locally convex space.
Let $H_{\tau}:=H^{\tau_1}(\mathbb{R}^n,\tau_2(x)dx)$ the weighted Sobolev Hilbert space with $\tau_1 \in \mathbb{N}, \tau_2(x)\in C^\infty(\mathbb{R}^n)$ and with $\tau_2 \ge 1$.
We denote $\mathcal{D}(\mathbb{R}^n)=\bigcap_\tau H_\tau$, when $\tau$ is arbitrary with stated properties above (and call $T$ the index set of all these $\tau$), endowed with the projective topology with respect to all the embeddings $\iota_\tau: \mathcal{D}(\mathbb{R}^n) \longrightarrow H_\tau$, i.e. the coarsest topology for which all these embeddings are continuous. This space is a nuclear and separable. The folowing construction will be the case, when $n=1$.
Also consider the symmetrical Fock spaces
$$\mathcal{F}(H_\tau):=\bigoplus^\infty_{n=0}H_{\tau, \mathbb{C}}^{\hat \otimes n}$$
of nth-symmetrized Hilbert space tensor products of the Sobolev spaces introduced above. This is the direct sum in of a countable collection of Hilbert spaces. We now consider the weighted Fock spaces
$$\mathcal{F}(H_\tau,p)=\lbrace (f_n) \in \mathcal{F}(H_\tau):\sum_n \|f_n\|^2_{H_{\tau,\mathbb{C}}^{\hat \otimes n}}\cdot p_n < \infty \rbrace,$$
for sequences $(p_n)$ with $p_n\ge 1$ for all $n\in \mathbb{N}$.
The space I am interested in is
$$\mathcal{F}_{\text{fin}}(\mathcal{D})=\bigcap\limits_{\tau \in T, p\ge 1 } \mathcal{F}(H_\tau,p)$$
the locally convex space as the projective limit of the weighted Fock spaces endowed with the projective topology, i.e. the initial topology with respect to all the embeddings $\iota_{\tau, p}:\mathcal{F}_{\text{fin}}(\mathcal{D}) \longrightarrow \mathcal{F}(H_\tau,p)$. I am wondering why this space is separable..
The separability is necessary to apply the so called projection spectral theorem in its most general form see [Y.M. Berezansky et al. Spectral Methods in Infinite-Dimensional Analysis, Chapter 3, Theorem 2.7]. This space gets used in the paper [Yu. M. Berezansky and V. A. Tesko. “The investigation of Bogoliubov func- tionals by operator methods of moment problem”. In: Methods Funct. Anal. Topology 22.1 (2016), pp. 1–47. issn: 1029-3531] as if it is separable, but the author does not specify why this is the case.
My current thoughts lead to the following:
If I can show, that
$$\bigcap_{\tau \in T}(H^{\tau_1}(\mathbb{R},\tau_2(x)dx))^{\otimes n}=\bigcap_{\tau \in T}H^{\tau_1}(\mathbb{R}^n,\tau_2^{\otimes n}(x)dx)=\mathcal{D}(\mathbb{R}^n),$$
then $\mathcal{F}_{\text{fin}}(\mathcal{D})=\bigoplus_k \mathcal{D}(\mathbb{R}^k)_{\text{symm},\mathbb{C}}$ as a set, i.e. the set of finite sequences. And since $\mathcal{D}(\mathbb{R}^k)$ is separable for all $k \in \mathbb{N}$, we can conclude $$\bigoplus_k \mathcal{D}(\mathbb{R}^k)_{\text{symm},\mathbb{C}}$$ is separable as a topological sum. Now If we additionally show that the topology of the sum is finer, than the topology of the projective limit, we are done.
Essential is the definition of weighted Sobolev space $H_\tau = H^{\tau_1}(\mathbb{R}, \tau_2)$ as the completion of $\mathscr{C}^\infty_0$ with respect to the following norm: $$ u \mapsto \left\| u \right\|^2_{\tau_1, \tau_2} = \sum_{|\alpha| \le \tau_1}\int \tau_2 \partial^\alpha u(x)\overline{\partial^\alpha u(x)}\: dx $$ Where $\mathscr{C}^\infty_0$ is the space of real-valued infinitely differentiable functions with compact support. $\mathscr{C}^\infty_0$ is given inductive topology. $\mathcal{D}(\mathbb{R})$ is the same space with projective topology. Both topologies are nuclear although. $\left\| \cdot \right\|_{\tau_1, \tau_2}$ norm is continuous for topology of $\mathscr{C}^\infty_0.$
It is not evident from the beginning that other definitions of weighted Sobolev spaces guarantees density of $\mathscr{C}^\infty_0.$ Common proof for unweighted case needs Holder-type equality for convolution (to apply mollification) that cannot be generalized to weighted case in a straight forward case. For the case of bounded regions, it is definitely true that $\mathscr{C}^\infty_0$ are not dense in any (even unweighted) Sobolev space: compare $\mathcal{H}$ vs. $\mathcal{H}_0.$
To prove separability of $\bigcap\limits_{\tau \in T, p\ge 1 } \mathcal{F}(H_\tau,p),$ it is sufficient to note that the map $$ \otimes^n\mathscr{C}^\infty_0(\mathbb{R}) \to \mathscr{C}^\infty_0(\mathbb{R}^n)\to \otimes^n \mathcal{H}^{\tau_1}(\mathbb{R}, \tau_2) $$ is continuous with dense image for any $\tau = (\tau_1, \tau_2).$ It is by definition for $n=1$ and by the general result: "if $H' \subset H$ is linear dense subspace of Hilbert space $H$ then $H'\otimes H'$ is dense in $H\widehat{\otimes }H$ and thus in $H \otimes H.$"
$$ \otimes^n\mathscr{C}^\infty_0(\mathbb{R}) \to \mathscr{C}^\infty_0(\mathbb{R}^n) \to \otimes_{sym}^n \mathcal{H}_\tau $$ is continuous with dense image, because symmetrization is continuous map (projection). $$ \oplus^{lc}_n\otimes^n\mathscr{C}^\infty_0(\mathbb{R}) \to \oplus^{lc}_n\otimes_{sym}^n \mathcal{H}_\tau $$ is continuous by universal property of sum applied to $\oplus^{lc}_n\otimes^n\mathscr{C}^\infty_0(\mathbb{R}).$ Topology of $\oplus^{lc}_n$ is that of inductive limit (final topology). Neighborhoods of 0 of $\oplus^{lc}$ have base, which are convex hulls $\Gamma(\oplus_n V_n)$ of sets of symmetric convex neighborhoods $V_n$ of 0 in each component $\otimes_{sym}^n \mathcal{H}_\tau.$
The image of the last map is dense. Take $(v_n)\in \oplus^{lc}_n\otimes_{sym}^n \mathcal{H}_\tau.$ It has finite-number of non-zero components, assume they have indices $ \le N.$ Neighborhood of $(v_n)$ is $(v_n) + U$ where $U$ is $\Gamma(\oplus_n V_n).$ Approximate $v_n, n \le N$ in $v_n + V_n$ with $d_n\in \otimes^n\mathscr{C}^\infty_0(\mathbb{R}), n \le N,$ take $d_n = 0, n > N,$ $(d_n) \in \oplus^{lc}_n\otimes^n\mathscr{C}^\infty_0(\mathbb{R})$ and $(d_n) \in (v_n) + U.$
Construct $$ \oplus_n\otimes^n\mathscr{C}^\infty_0(\mathbb{R}) \to \oplus^{lc}_n\otimes_{sym}^n \mathcal{H}_\tau \to \oplus^{Hilb, p}_n\otimes_{sym}^n \mathcal{H}_\tau $$ by final property of $\oplus^{lc}$ applied to $\otimes_{sym}^n \mathcal{H}_\tau.$ $\oplus^{Hilb, p}$ is generated by norms $p_n \left\| \cdot \right\|_n$ in each component. This is continuous map. It has dense image as topology is weakened.
The following map is continuous: $$ \oplus_n\otimes^n\mathscr{C}^\infty_0(\mathbb{R}) \to \widehat{\oplus}^{Hilb, p}_n\otimes_{sym}^n \mathcal{H}_\tau $$ ($\widehat{\oplus}$ denotes completion) and the image is dense, because the space is dense in its completion.
By universal property, the following map is also continuous: $$ \oplus_n\otimes^n\mathscr{C}^\infty_0(\mathbb{R}) \to \cap_{p, \tau}\widehat{\oplus}^{Hilb, p}_n\otimes_{sym}^n \mathcal{H}_\tau $$ Sets \begin{align} LH &= \cap_{\tau, p}\widehat{\oplus}^{Hilb, p}_n\otimes_{sym}^n \mathcal{H}_\tau \\ L &= \cap_{\tau}\oplus^{lc}_n\otimes_{sym}^n \mathcal{H}_\tau \end{align} are equal due to arbitrary $p$ (topologies are different).
Base of topology of $LH$ consists of finite intersections of $\eta^{\prime -1}_{\tau_i, p_i}(\Omega_{\tau_i, p_i})$ where $\Omega_{\tau_i, p_i} \subseteq \widehat{\oplus}^{Hilb, {p_i}}\otimes_{sym}^n \mathcal{H}_{\tau_i}$ open and $$ \eta'_{\tau, q}: LH \to \widehat{\oplus}^{Hilb, q}_n\otimes_{sym}^n \mathcal{H}_\tau $$ are canonical (initial topology in category of l.c. topological vector spaces and in topological vector spaces sets are the same). As $\eta'_{\tau, q}$ maps to $\oplus^{Hilb, q}_n\otimes_{sym}^n \mathcal{H}_\tau$ and topology on $\oplus^{Hilb, q}_n\otimes_{sym}^n \mathcal{H}_\tau$ is induced from completion, one can consider: $$ \eta_{\tau, q}: LH \to \oplus^{Hilb, q}_n\otimes_{sym}^n \mathcal{H}_\tau $$ and $$ LH = \cap_{\tau, p}\widehat{\oplus}^{Hilb, p}_n\otimes_{sym}^n \mathcal{H}_\tau = \cap_{\tau,p}\oplus^{Hilb, p}_n\otimes_{sym}^n \mathcal{H}_\tau $$ as l.c. topological vector spaces with initial topology induced by maps: $$ \eta_{\tau, q}: LH \to \oplus^{Hilb, q}_n\otimes_{sym}^n \mathcal{H}_\tau $$
Obtained continuous map: $$ \oplus_n\otimes^n\mathscr{C}^\infty_0(\mathbb{R})\to \cap_{\tau, p}\oplus^{Hilb, p}_n\otimes_{sym}^n \mathcal{H}_\tau $$ with dense image noting that topology is described by finite intersection, see above for $\eta^{\prime -1}$ and use $\eta^{-1}$ instead.
$\oplus_n\otimes^n\mathscr{C}^\infty_0(\mathbb{R})$ has dense countable subset by post
[Yu. M. Berezansky and V. A. Tesko. “The investigation of Bogoliubov functionals by operator methods of moment problem”] refers to monograph of [Yu. M. Berezansky, Z. G. Sheftel, G. F. Us, Functional Analysis, Vols. 2, ch. 14] for the complete proof of the result. It looks like that monograph is a strong prerequisite to read the paper :)