Approximating dense subspaces of Fréchet spaces

Here is a very simple method for separable Hilbert spaces (which easily generalizes to Frechet spaces with Schauder bases): Take an orthonormal basis $(e_k)_k$ in $H_0$ and choose $f_{n,k}\in H_1$ such that $\|e_k-f_{n,k}\|_0 \le 1/(n^2+k^2)$. Then define $P_n(x)=\sum_{k=1}^n \langle e_k,x\rangle_0 f_{n,k}$. The difference $\|P_n(x)-x\|_0$ is estimated just using the triangle inequality (so that this construction isn't bound to Hilbert spaces).

EDIT. I add some details for the Frechet case showing (at least under a mild additional assumption) that one does not need an absolute basis (which would exclude many Banach spaces, I recall the definitions below). Let $(\|\cdot\|_N)_{N\in\mathbb N}$ be an increasing sequence of seminorms giving the topology of the Frechet space $H_0$. A Schauder basis $(e_k)_k$ is a sequence such that that every $x\in H_0$ has a unique representation $x=\sum\limits_{k=1}^\infty \xi_k(x)e_k$. (The basis is called absolute if, for each $N$, there are $M$ and $c>0$ such that $\sum\limits_{k=1}^\infty |\xi_k(x)|\|e_k\|_N \le c\|x\|_M$ for all $x\in H_0$ -- this implies that the spaces is a projective limit of weighted $\ell^1$ spaces and excludes Hilbert spaces). By corollary 28.11 in the book Introduction to Functional Analysis of Meise and Vogt one has a slightly weaker condition for every Schauder basis: For every $N$ there are $M=M(N)$ and $c>0$ such that $\sup\lbrace|\xi_k(x)|\|e_k\|_N:k\in\mathbb N\rbrace \le c\|x\|_M$. In particular, the coefficient functionals $\xi_k$ (which are linear by the uniqueness) are continuous.

We construct $P_n$ under the additional assumptions that $\|\cdot\|_1$ is a norm and not only a seminorm (I am quite optimistic that this can be removed). For $n,k \in\mathbb N$ choose $f_{n,k}\in H_1$ with $\|e_k-f_{n,k}\|_n\le \|e_k\|_1/n^2$ and set, as previously, $P_n(x)=\sum\limits_{k=1}^n\xi_k(x)f_{n,k}$. These are continuous linear operators $H_0\to H_1$, and for each $x\in H_0$, $N\in\mathbb N$, and $n\ge N$ we have $$ \|P_n(x)-x\|_N \le \sum_{k=1}^n |\xi_k(x)|\|f_{n,k}-e_k\|_N + \|\sum_{k=n+1}^\infty\xi_k(x)e_k\|_N. $$ The second term tends to $0$ and (since $n\ge N$) the first term can be estimated by $\sum_{k=1}^n |\xi_k(x)|\|e_k\|_1/n^2 \le c\|x\|_{M(1)}/n$.

Thanks to Jochen, Matthew and Bill, this is a detailed proof for Fréchet spaces.

Proposition. Let $E$ be a separable Fréchet space with the bounded approximation property, $F$ a topological vector space, continuously and densely embedded in $E$. Then there exists a sequence of continuous linear maps $P_n : E \to F$, such that $$ \forall x \in E\,:\, P_n x \to x \text{ in }E\,. $$

Proof. Let $(x_n)_{n \in \mathbb N}$ be a countable, dense sequence in $E$ and $(\|\cdot\|_n)_{n \in \mathbb N}$ an increasing fundamental system of seminorms. We assumed that $E$ has the bounded approximation property, hence there exists an equicontinuous sequence of linear maps $T_n : E \to E$ with finite rank that converge to $\operatorname{Id}_E$, uniformly on compact sets. By passing to a subsequence we can assume that $$ \| T_n x_j - x_j \|_n \leq \frac 1n \text{ for }j \leq n\,. $$ Due to equicontinuity there exists for each $m$, an $N_m \in \mathbb N$ and $C_m>0$ such that $$ \forall n \in \mathbb N\,,\; \forall x \in E\,:\, \| T_n x \|_m \leq C_m \| x \|_{N_m}\,. $$

For each $n$, the space $T_n(E)$ is finite dimensional. Let $n'=n'(n)$ be such that $\|\cdot\|_{n'}$ is a norm on $T_n(E)$. We can construct a map $S_n : T_n(E) \to F$ with $$ \| S_ny - y \|_{n} \leq \frac 1n \| y \|_{n'}\,, $$ for all $y \in T_n(E)$. To see that this is possible choose a basis $y_1, \dots, y_m$ of $T_n(E)$ and note that it is sufficient to define $S_n(y_i) \in F$, such that $\| S_n(y_i) - y_i \|_{n}$ is small enough. This is possible, because $F$ is dense in $E$. Define $P_n = S_n T_n$.

We have to show convergence $P_n x \to x$. Fix $x \in E$ and a seminorm $\|\cdot\|_m$. For $n$ and $k$ satisfying $m ,\,N_m,\,N_{m'} \leq n$ and $k \leq n$ we have \begin{align*} \| P_n x &- x \|_m \leq \|S_n T_n(x-x_k) -T_n(x-x_k) \|_m + \| T_n(x-x_k) \|_m + \\ &\qquad\qquad\qquad + \|S_n T_n x_k - T_n x_k \|_m +\| T_n x_k - x_k\|_m + \| x_k - x\|_m \\ &\leq \frac 1n \| T_n(x-x_k) \|_{m'} + C_m \| x - x_k \|_{N_m} + \frac 1n \| T_n x_k \|_{m'} + \frac 1n + \|x_k - x \|_m \\ &\leq \frac {C_{m'}}{n} \| x-x_k \|_{N_{m'}} + C_m \| x - x_k \|_{N_m} + \frac {C_{m'}}n \left( \| x\|_{N_{m'}} + \| x-x_k \|_{N_{m'}} \right) + \\ &\qquad\qquad + \frac 1n + \|x_k - x \|_m \,. \end{align*} We see that by choosing $n$ large enough and $\|x - x_k\|_n$ small enough we can achieve convergence.