Let $(X_s)_{s \in (0,s_1)}$ be an increasing sequence of Banach spaces with the property that if $0<s<r<s_1$, then
$$ \|u\|_{X_s} \leq \|u\|_{X_r}. $$
We define
$$\tilde{X}_s = \projlim_{r>s} {X_r}.$$
Please help me understand the spaces $\tilde{X}_s$.
Context: If it helps, the context of my question is PDE. The spaces $X_s$ are actually function spaces such that a differential operator brings an element of $X_s$ to a larger space $X_r$.
At the top of my head, the questions I have are:
- What are the elements of $\tilde{X}_s$?
- What kind of a space is this? I think I have read somewhere that $\tilde{X}_s$ will turn out to be a Frechet space. Is this correct?
- If I apply a differential operator to an element of $\tilde{X}_s$, what happens?
Note: I have already looked at some of the answers here regarding projective limit or inverse limit but all of them were too advanced for me to understand. If I have missed an answer that might be helpful in my case, feel free to give me the link instead. Thanks!
$\tilde X_s = \bigcap\limits_{r>s} X_r = \bigcap\limits_{n\in\mathbb N} X_{s+1/n}$ is endowed with the (locally convex) topology having all $\|\cdot\|_r$, $r>s$, as seminorms (or only $\|\cdot\|_{s+1/n}$, $n\in\mathbb N$). Then convergence $x_\alpha \to x$ in $\tilde X_s$ is equivalent to $x_\alpha \to x$ in all $X_r$ (or only in $X_{s+1/n}$). $\tilde X_s$ is indeed a Fréchet space: It is metrizable because the topology can be given by countably many seminorms and completeness follows rather easily from the completeness of all $X_r$.
If you have a linear operator $T$ on some $X_t$ for $t>s$ you can restrict it to the intersection. Then it will be continuous on $\tilde X_s$ if and onlyif, for every $p\in (s,t)$ there is $q\in (s,t)$ such that $T:X_q\to X_p$ is continuous.