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.



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 
$\sum\_{k=1}^n |\xi\_k(x)|\|e\_k\|\_1/n^2 \le c\|x\|\_{M(1)}/n$.


