The standard (or not so standard, as analysis books tend to give messy constructions instead) conceptual way to define the topology is very well explained in Bourbaki's topological vectors spaces.
In general: Let $(E_i,f_{ij})$ be an inductive system of vector spaces (over $\mathbf{C}$, say) such that every $E_i$ is endowed with an LC topology making the maps $f_{ij}:E_i\rightarrow E_j$ continuous. If $E:=\varinjlim E_i$ denotes its inductive limit in $\mathbf{C}-\mathbf{Vect}$, then $E$ is said to be the locally convex inductive limit of $(E_i,f_{ij})$ (inductive limit in $\mathbf{LCS}$) if it is given the final locally convex topology with respect to the family of canonical maps $f_i:E_i\rightarrow E$.
The following is a special case: Consider a countable increasing family $E_1\subset E_2\subset\cdots$ of vector subspaces of a vector space $E$ such that $E=\bigcup_{i\in\mathbf{N}} E_i$ and every $E_i$ be endowed with a locally convex topology such that for every $i\in\mathbf{N}$ the subspace topology induced by $E_{i+1}$ on $E_i$ coincides with the topology of $E_i$. Then $(E_i,f_{ij})$ is an inductive system with respect to the inclusion maps $f_{ij}:E_i\hookrightarrow E_j$ ($i\leqslant j$), and $E$ is its strict locally convex inductive limit. If every $E_i$ is a Fréchet space, then $E$ is said to be an LF space.
In particular, let $\Omega$ be an open subset of $\mathbf{R}^n$; if $K$ is a compact subset of $\Omega$, let $\mathcal{D}_K(\Omega)$ be the subset of $C^\infty_0(\Omega)$ of maps with support contained in $K$. $\mathcal{D}_K(\Omega)$ is a Fréchet space with the topology given by uniform convergence of all derivatives; a generating family of seminorms is given by $\rho_k(\phi):=\sup_{x\in\Omega, |\alpha|\leqslant k}|(\partial^\alpha\phi)(x)|$. Let $K_1\subset K_2\subset\cdots$ be an exhaustion of $\Omega$ by compact subsets. Then $C^\infty_0(\Omega)=\bigcup_{i\in\mathbf{N}}\mathcal{D}_{K_{i}}(\Omega)$ and let $\mathcal{D}(\Omega):=\varinjlim\mathcal{D}_{K_{i}}(\Omega)$ be its strict locally convex inductive limit.
I'm going to consider a part of your Question 1, namely:
What is the reason most people don't bother talking about the actual topology and seems satisfied with sequences, although the topology is not sequential?
I think that there are (at least) two reasons for this. The first is technical:
- The topology is not easy to define and it is not easy to manipulate (here "not easy" means "not easy for an introductory course", for example a course with focus in the applications to PDE).
The second is more relevant:
- The topology doesn't matter for the basic properties of distributions (probably, for the topics of the said introductory courses in which the said topology is not defined).
Sounds unsatisfactory, right? I agree, so let me explain. These are words (not literally) of Laurent Schwartz, who created the theory of distributions. In fact, Schwartz said the following with respect to the time in which he started work with the test functions:
I was unable to put a topology on $\mathcal{D}$, but only what I called a pseudo-topology, i.e. a sequence $(\phi_n)$ converges to $0$ in $\mathcal{D}$ if the $(\phi_n)$ and all their derivatives converge uniformly to $0$, keeping all their supports in a fixed compact set. I only found an adequate topology much later, in Nancy in 1946. But it doesn't matter for the main properties. ([1], p. 229-230).
This quote teach us the following:
- Historically, the notion of convergence of sequences in $\mathcal{D}$ came before the topology of $\mathcal{D}$.
As a consequence, it is natural to begin the study of distribution theory with the notion of convergence (instead of start with the actual topology).
In addition, the quote draw our attention for the following fact:
- There are problems that you can solve in the context of distributions without invoke a topology for $\mathcal{D}$. For some purposes, the usual notion of convergence (which Schwartz called pseudo-topology) is enough.
For example, the fact that the distributional derivative "preserves convergence of sequences" (in $\mathcal{D}'$) is a result that can be obtained and applied to the differential equations without the actual topology of $\mathcal{D}$.
Remark: Sometimes this result is called "continuity" of the distributional derivative, even in the context where the notion of convergence in $\mathcal{D}'$ is defined as the convergence in $\mathcal{D}$: the explicit form of the convergence is given but a topology is not defined. However, it is indeed possible to put a topology on $\mathcal{D}'$ (which implies the said notion of convergence in $\mathcal{D}'$) without put a topology on $\mathcal{D}$. With respect to this topology in $\mathcal{D}'$ the distributional derivative is indeed "continuous" (and thus preserves convergence of sequences). To give a reference for this remark, let me quote what Schwartz said in his treatise:
Nous définissons ainsi sur $\mathcal{D}'$ une topologie (qui, remarquons-le encore, ne nécessite pas la connaissance de la topologie de $\mathcal{D}$, mais seulement de ses ensembles bornés). ([2], p. 71)
Of course, as the quote suggests, Schwartz could define boundedness in $\mathcal{D}$ even in absence of a topology:
I did not have a topology on $\mathcal{D}$, but what I called a pseudo-topology [...]. I could speak without difficulty of a bounded subset of $\mathcal{D}$ [...]. $\mathcal{D}$ was more or less one of the spaces I had studied deeply during that short period [summer of 1943], always with the slight difficulty of the pseudo-topology, which nevertheless did not stop me. ([1], p. 231)
In short, all these things support the fact that is it possible to do (and Schwartz certainly did) many things in the context of the distributions without appeal to the topology of $\mathcal{D}$ (but only with the notion of convergence). In my opinion this justifies the second reason above as a fundamental answer for your "why". Maybe we could just say that people avoid talking about the topology (in some contexts) because it is an efficient strategy (in the context where it is avoided). The point is that the topology was created to yields a prior notion of convergence and allow a deeper development of the theory. The notion of convergence is not a mere simplification to avoid a complicated topology whose origin is a mystery; of course the topology is complicated and people make it seems mysterious (by virtue of an explanation's lack), but the notion of convergence is the cause of the topology and not the converse. Maybe you will agree that, from this point of view, the fact that in some contexts "people don't bother talking about the actual topology" becomes natural and acceptable.
Addendum (details on the creation of the topology). What was the advantage of defining a topology on $\mathcal{D}$? It was to make possible the application of the knows theorems of topological spaces, like the Hanh-Banach Theorem. The last sentence seems vague and sounds like a cliche, right? But it is the truth; it was essentially what Schwartz said:
In Grenoble, I gave an exact definition of the real topology corresponding to the pseudo-topology on $\mathcal{D}$, which later, in 1946, Dieudonne and I took to calling an inductive limit topology. The pseudo-topology is not enough; in order to apply the Hahn-Banach theorem and to study the subspaces of $\mathcal{D}$, you need to work with a real topology. ([1], p. 238)
I carefully defined the neighborhoods of the origin in $\mathcal{D}$, then gave the characteristic property which was precisely that of being an inductive limit, without giving it a name. I only did this for the particular object $\mathcal{D}$, without daring to introduce a general category of objects. Mathematical discovery often takes place in this way. One hesitates to introduce a new class of objects because one needs only one particular one, and one hesitates even more before naming it. It's only later, when the same procedure has to be repeated, that one introduces a class and a name, and then mathematics takes a step forwards. Other inductive limits were
introduced, then the theory of sheaves used them massively and homological algebra showed the symmetry of inductive and projective limits. ([1], p. 283)
[1] A Mathematician Grappling with His Century by Laurent Schwartz.
[2] Théorie des distributions by Laurent Schwartz.
Best Answer
I’m not sure about your conventions, so let’s say $(\tau_h\phi)(x)=\phi(x-h)$, so that the result is correct.
Let $F_h=\frac{\phi-\tau_h\phi}{h}-\phi’$ for $h \neq 0$.
We want to show that in $\mathcal{D}(\mathbb{R})$, $F_h \rightarrow 0$ as $h \rightarrow 0$. This means showing two things:
that there exists a compact subset $K \subset \mathbb{R}$ and $h_0>0$ such that for each $h$ such that $0<|h| < h_0$, $F_h$ has support in $K$.
that for every $n \geq 0$, $\sup_{x \in \mathbb{R}}\,|F_h^{(n)}(x)| \rightarrow 0$ as $h \rightarrow 0$.
For 1), just take $K=L+[-1,1]$ where $L$ is the support of $\phi$ (since $\phi \in \mathcal{D}(\mathbb{R})$) and $h_0=1$.
Let’s see 2). For each $n \geq 0$, for each real number $x$, for each $h \neq 0$, $F_h^{(n)}(x)=\frac{\phi^{(n)}(x)-\phi^{(n)}(x-h)-h\phi^{(n+1)}(x)}{h}=-h^{-1}(\phi^{(n)}(x-h)-\phi^{(n)}(x)-(-h)\phi^{(n+1)}(x))$, so by Taylor, $$F_h^{(n)}(x)=-h^{-1}\int_x^{x-h}{(x-h-t)\phi^{(n+2)}(t)dt}=-h^{-1}\int_0^{-h}{(-t-h)\phi^{(n+2)}(x+t)dt},$$ thus $$F_h^{(n)}(t)=h^{-1}\int_h^0{t\phi^{(n+2)}(x-h+t)dt}=-h\int_0^1{t\phi^{(n+2)}(x-h+ht)dt}.$$
Finally, $|F_h^{(n)}(x)| \leq |h| \|\phi^{(n+2)}\|_{\infty}$, hence the conclusion.