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.