What is a decreasing scale of Banach spaces?

Question

I am a having a hard time understanding a part of an article I am reading. The screen-cap is below. Basically, it's the line labeled (6) that I do not understand. I am not familiar with the circular arrow notation. From what I can guess, it represents that there is some "embedding" from $F_r(\Omega)$ to $F_s(\Omega)$. But what kind of embedding is this? (It seems there are different kinds of embedding and I'm really only familiar with topological embeddings.)

This is what I can work out so far: For $0<s<r$, we have that $\Omega_s \subseteq \Omega_r$ (same inclusion holds for their closures). Thus, if $f \in F_r(\Omega)$, we have that $f$ is also in $F_s(\Omega)$. So in this sense, $(F_s(\Omega))_{s>0}$ is a decreasing sequence of spaces. More precisely, I can define $\gamma: F_r(\Omega) \to F_s(\Omega)$ by $f \mapsto f$. Clearly, the range $R$ of $\gamma$ contains the set of analytic functions, and so the closure $\bar{R}$ of $R$ in $F_s(\Omega)$ is $F_s(\Omega)$ itself. I guess this is what it means to have dense range?

So now, what exactly is a decreasing scale of Banach spaces and what does "embedding of norm 1" mean?

Answer 1

The circular arrow notation is an attempt of a math-challenged typesetter to represent the inclusion sign $\subset$ with an arrow on the bottom end. These days one usually types \hookrightarrow in $\LaTeX$, as in $F_r(\Omega)\hookrightarrow F_s(\Omega)$.

In many contexts, embedding means "a map that is an isomorphism onto its range", where the notion of isomorphism is taken from the context. For example, a map $F:X\to Y$ is a topological embedding if it is a homeomorphism between $X$ and $F(X)$. To give another example: in Banach space theory, an isomorphism means an invertible linear operator. Therefore, one says that $f:X\to Y$ is an embedding of Banach space $X$ into a Banach space $Y$ if $F$ is an invertible linear operator between $X$ and $F(X)$; equivalently, if there exist positive finite constants $c$ and $C$ such that $$c\|x\|_X\le \|F(x)\|_Y\le C\|x\|_X \tag1$$

However, people working with spaces of functions have their own way of using the term embedding. They say that a function space $X$ embeds into function space $Y$ if every function $\phi\in X$ also belongs to $Y$ and
$$\|\phi\|_Y\le C\|\phi\|_X \tag2$$ for some constant $C$ independent of $\phi$. The best (smallest) possible of $C$ in (2) is called the embedding constant, or the norm of the embedding. This is the notion of embedding used in your source.

In your example, the norm of function $f$ is the supremum of $|f|$ over some set. When $s<r$, the supremum over $\Omega_s$ cannot be larger than the supremum over $\Omega_r$. This translates into $$ \|f\|_{E_s(\Omega)}\le \|f\|_{E_r(\Omega)} \tag3 $$ which is (2) with $C=1$. Also, $1$ is the best possible constant here because for $f\equiv 1$ we have equality in (3). This is why the norm of the embedding is $1$.

Having a dense range indeed means that the range is dense. As the author says, the definition of $F_s$ is made so that the density of range is easy to show: it's basically a tautology. For the original space $E_s$ the density would not be as easy to prove, if it's true at all.

Summary:

"embedding of norm 1" means that (2) holds with best possible constant $C=1$.

"embedding with dense range" means that the range of the embedding is a dense subset of the codomain.

"decreasing scale of Banach spaces" is defined in the last sentence you quoted: it is family of Banach spaces $B_t$ such that for $t<s$, $B_s$ embeds into $B_t$ (in the sense (2)) with norm $1$ and dense range.

In general, writing "$X$ has properties 1,2,3 ($X$ is a widget)" is an (arguably, suboptimal) way to simultaneously define widget as an object with properties 1,2,3, and to assert that $X$ has those properties.