We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support
Example: For instance bump function is in $\mathcal{D}(\mathbb R)$
Let $E$ is a Banach function space on $\mathbb R.$
We put, $E^{\mathrm{loc}}=$ The collections of all functions $f$ such that $\phi f \in E$ for all $\phi \in \mathcal{D}(\mathbb R);$ that is, $E^{\mathrm{loc}}=\{f:\mathbb R\to \mathbb C: \phi f \in E, \forall \phi \in \mathcal{D}(\mathbb R) \}.$
Example: (a) $E=L^{1}(\mathbb R)$, (b) $E=H^{s}(\mathbb R), (s>\frac{1}{2})$ (The Sobolev space); are well-known Banach spaces
My naive Questions are:
(1) If $E= L^{1}(\mathbb R)$ or $E=H^{s}(\mathbb R);$ then what is $E^{\mathrm{loc}}$ intuitively ?
(2) What are the ideas behind considering the (bit roughly speaking) local space $E^{\mathrm{loc}}$ for the given Banach space ? How to justify the word locally ?
(3) What's the underneath institutive thoughts for such consideration and if possible, can you show me, its importance, some where in application(of course, I mean somewhere in Mathematics itself)?
Thanks,
Forming the convolution of the (scaled) bump function with indicator functions you get "qausi-indicator functions" in $\mathscr D$, in particular, there are $\psi_n\in\mathscr D(\mathbb R)$ which are positive and equl to $1$ on $[-n,n]$. It is then easy to see that the elements of $E^{loc}$ are those distributions which, on every compact set, have the same behavior as the elements of $E$ (in particular, if the elements of $E$ are functions the same is true for $E^{loc}$). For instance, $L^1(\mathbb R)^{loc}$ is the space (of equivalence classes w.r.t. equality almost everywhere) of functions $f$ such that $\int_K |f(x)|dx <\infty$ for all compact $K$, that is the locally integrable functions.
You can see these local spaces at work in the second volume of H\"ormander's Analysis of Partial Differential Operators.