General method to prove density, continuous and compact embedding of space into another

Question

We say that a set $X$ is dense into another one $X'$ if for any $x$ $\in$ $X'$ there exists a sequence $x_n$ that is in $X$ such that $$\lim\limits_{n\to \infty}x_n=x$$ we say that a set $X$ is compactly embedded into $Y$ if from any uniformly bounded sequence $x_n$ of $X$ one can extract a subsequence $x_{\varphi(n)}$ that converges in $Y$. Finally a set $X$ is said to be continuously embedded into $Y$ if $$\|x\|_Y\leq \|x\|_X$$ whenever $x$ belongs to $X$. Now, the question is: is there any general method or even a set of several methods to postulate whether a space is dense into another or not, continuously embedded or not compactly embedded or not?

Answer 1

One typical pattern of events is that we start with a "small" space $D$ (e.g., test functions), and consider its completions $H^s$ with respect to norms $|\cdot|_s$, where $|\cdot|_s\le |\cdot|_t$ for $s<t$. Then $D$ is dense in every $H^s$, by construction, $H^t$ imbeds continuously in $H^s$ for $t>s$, and since $D$ is dense in $H^s$, certainly $H^t$ is dense in $H^s$.

Proof of compactness of $H^t\to H^s$ is more delicate, and is less widely true than the previous. Often, when it does hold, such as with the Hilbert-space Sobolev spaces on the circle, it's because $H^s$ and $H^t$ have a common orthogonal basis (e.g., exponentials), and the identity map can be examined explicitly as a map $\ell^2\to \ell^2$ by multiplication by a sequence of reals $\mu_n$. When the $\mu_n$'s go to $0$, the map is compact.