TITLE: Approximating dense subspaces of Fréchet spaces
QUESTION [6 upvotes]: If $H$, $H_0$ are two separable Hilbert spaces and $H$ is continuously and densly embedded in $H_0$, it is possible to construct a sequence of linear operators
$$ P_n : H_0 \to H $$
such that for all $x \in H_0$ one has convergence $P_n x \to x$ in the $H_0$-norm.
The motivation is to generalize the idea of smoothing operators. For example, $H^2(\mathbb R)$ is densely embedded in $L^2(\mathbb R)$ and it is well known that one can approximate each $L^2$-function by an $H^2$-function. In this case $P_n$ could be a convolution with a smooth kernel with width $1/n$.
Can this be generalized to Fréchet spaces or even beyond? If $H$ and $H_0$ are Fréchet spaces rather than Hilbert spaces, is it always possible to construct a sequence of bounded operators $P_n$ as above?
In the Hilbert case one way to construct $P_n$ is by considering an unbounded, self-adjoint operator $A : D(A) \to H_0$, with $D(A)=H$, that represents the inner product via
$$ \langle v,w \rangle_H = \langle Av, Aw \rangle_{H_0}\,. $$
If $\{ P_\Omega\}$ is the spectral measure accociated to $A$, then we can use $P_{[-n,n]}$ as our sequence of operators. It is less clear to me, what to with Fréchet spaces.

REPLY [3 votes]: 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.