# Doubt in understanding Space $\mathscr D(\Omega)$

I was reading about distributions from Rudin. I had 2 doubts in understanding space $$\mathscr D(\Omega)$$. Here is the relevant section:

6.2 The space $$\mathscr{D}(\Omega)$$ Consider a nonempty open set $$\Omega \subset R^{n}$$. For each compact $$K \subset \Omega$$, the Fréchet space $$\mathscr{D}_{K}$$ was described in Section 1.46. The union of the spaces $$\mathscr{D}_{K}$$, as $$K$$ ranges over all compact subsets of $$\Omega$$, is the test function space $$\mathscr{D}(\Omega)$$. It is clear that $$\mathscr{D}(\Omega)$$ is a vector space, with respect to the usual definitions of addition and scalar multiplication of complex functions. Explicitly, $$\phi \in \mathscr{D}(\Omega)$$ if and only if $$\phi \in C^{\infty}(\Omega)$$ and the support of $$\phi$$ is a compact subset of $$\Omega$$. Let us introduce the norms $$\|\phi\|_{N}=\max \left\{\left|D^{\alpha} \phi(x)\right|: x \in \Omega,|\alpha| \leq N\right\}\tag1$$ for $$\phi \in \mathscr{D}(\Omega)$$ and $$N=0,1,2, \ldots$$; see Section $$1.46$$ for the notations $$D^{\alpha}$$ and $$|\alpha|$$.

The restrictions of these norms to any fixed $$\mathscr{D}_{K} \subset \mathscr{D}(\Omega)$$ induce the same topology on $$\mathscr{D}_{K}$$ as do the seminorms $$p_{N}$$ of Section $$1.46$$. To see this, note that to each $$K$$ corresponds an integer $$N_{0}$$ such that $$K \subset K_{N}$$ for all $$N \geq N_{0}$$. For these $$N,\|\phi\|_{N}=p_{N}(\phi)$$ if $$\phi \in \mathscr{D}_{K} .$$ Since $$\|\phi\|_{N} \leq\|\phi\|_{N+1} \quad \text { and } \quad p_{N}(\phi) \leq p_{N+1}(\phi)\tag2$$ the topologies induced by either sequence of seminorms are unchanged if we let $$N$$ start at $$N_{0}$$ rather than at $$1 .$$ These two topologies of $$\mathscr{D}_{K}$$ coincide therefore; a local base is formed by the sets $$V_{N}=\left\{\phi \in \mathscr{D}_{K}:\|\phi\|_{N}<\frac{1}{N}\right\} \quad(N=1,2,3, \ldots)\tag3$$ The same norms (1) can be used to define a locally convex metrizable topology on $$\mathscr{D}(\Omega)$$; see Theorem $$1.37$$ and $$(b)$$ of Section $$1.38$$. However, this topology has the disadvantage of not being complete. For example, take $$n=1, \Omega=R$$, pick $$\phi \in \mathscr{D}(R)$$ with support in $$[0,1], \phi>0$$ in $$(0,1)$$, and define $$\psi_{m}(x)=\phi(x-1)+\frac{1}{2} \phi(x-2)+\cdots+\frac{1}{m} \phi(x-m)$$ Then $$\left\{\psi_{m}\right\}$$ is a Cauchy sequence in the suggested topology of $$\mathscr{D}(R)$$, but $$\lim \psi_{m}$$ does not have compact support, hence is not in $$\mathscr{D}(R)$$.

(Trascribed from screenshots 1,2,3.)

Doubts:

1. Why are the topologies on $$\mathscr D(\Omega)$$ and $$\mathscr D_k$$ the same?

2. Why is $$\{\psi_m\}$$ a Cauchy sequnce but its limit doesn't have compact support?

I am studying functional analysis on my own with only the help of Math Stackexchange. Any help will be appreciated.

• 1) This is the definition : $U\subset D(\Omega)$ is open iff for all compact $K$, $U\cap D_K$ is open (in $D_K$). Jan 16, 2020 at 7:46
• Here is my advise: do not read Rudin. There are many other good books with nice motivations from the field of PDEs, Physics books, etc. Rudin is written in a Bourbakist style, useful only if you know the contents beforehand and you are looking for some concrete proof/fact. Jan 16, 2020 at 8:18
• @GReyes Thanks a lot, Sir. Please can you suggest some books. I am interested in field of pde. Jan 16, 2020 at 13:23
• I really love this one: V. S. Vladimirov (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029. Jan 16, 2020 at 16:08
• (I can't resist commenting that Rudin does not even manage to be "Bourbakist"... and he would have been offended at any such label. His mathematical culture was a different, mostly peculiarly U.S.-centric, much-more-informal version of Bourbaki's idea to "be careful"...) Feb 28, 2020 at 0:33

Here is a crash course on the topology of $$\mathcal{D}(\Omega)$$.

Let $$V$$ be a vector space over $$\mathbb{R}$$. I will restrict to real scalars but one can also treat in the same way vector spaces over $$\mathbb{C}$$. $$V$$ is called a topological vector space if it is equipped with a topology $$\mathscr{T}$$ such that $$+:V\times V\rightarrow V$$ and $$\cdot:\mathbb{R}\times V\rightarrow V$$ are continuous. Here $$V\times V$$ is given the product topology coming from $$\mathscr{T}$$ for each factor. Likewise $$\mathbb{R}\times V$$ is given the product topology of the usual topology of $$\mathbb{R}$$ and the topology $$\mathscr{T}$$ on $$V$$.

A map $$\rho:V\rightarrow \mathbb{R}$$ is called a seminorm on $$V$$ iff it satisfies the three conditions:

1. $$\forall v\in V, \rho(v)\ge 0$$
2. $$\forall v,w \in V, \rho(v+w)\le \rho(v)+\rho(w)$$
3. $$\forall v\in V, \forall \lambda\in\mathbb{R}, \rho(\lambda v)=|\lambda|\rho(v)$$

Let $$s(V)$$ denote the set of all seminorms on $$V$$. Given a subset $$A$$ of $$s(V)$$, one can define a topology $$\mathscr{T}_A$$ on $$V$$ as follows. First for $$v\in V$$, $$r>0$$ and $$\rho\in A$$, define the "open ball" $$B(v,r,\rho)=\{w\in V\ |\ \rho(w-v) Now let $$\mathscr{T}_A$$ be the smallest topology on $$V$$ which contains the set of all such open balls (i.e., use the collection of these balls as a subbasis for defining a topology). This makes $$V$$ into a topological vector space (TVS) [Exercise 1: prove this]. A TVS which can be obtained in this way is called a locally convex TVS (LCTVS) [Remark 1: you don't have to prove this, it's a definition].

A seminorm $$\eta$$ on a LCTVS $$V$$ is called a continuous seminorm iff it is continuous in the usual sense, i.e., as a map between the topological spaces $$V$$ and $$\mathbb{R}$$. If $$V$$ is given as above, starting from a set of defining seminorms $$A$$, then the latter property is equivalent to $$\exists k\ge 0, \exists \rho_1,\ldots,\rho_k\in A, \exists c_1,\ldots,c_k\ge 0, \forall v\in V,$$ $$\eta(v)\le c_1\rho_1(v)+\cdots+c_k\rho_k(v)\ .$$ [Exercise 2: prove this equivalence]

Let $$V_1,\ldots,V_n,W$$ be LCTVS's. Let $$\phi:V_1\times\cdots\times V_n\rightarrow W$$ be an $$n$$-linear map. Give $$V_1\times\cdots\times V_n$$ the product topology. Then $$\phi$$ is a continuous map iff for all continuous seminorm $$\eta$$ on $$W$$, there exist continuous seminorms $$\rho_1,\ldots,\rho_n$$ on $$V_1,\ldots,V_n$$ respectively, such that $$\forall v_1\in V_1,\ldots,\forall v_n\in V_n,\ \ \eta(\phi(v_1,\ldots,v_n))\le \rho_1(v_1)\cdots\rho_{n}(v_n)\ .$$ [Exercise 3: prove this last equivalence too]

Clearly, if the topology of $$W$$ is given as $$\mathscr{T}_A$$ for some $$A\subset s(W)$$, it is enough to check the last condition for $$\eta$$'s in $$A$$ only.

Example 1: Let $$\Omega$$ be a nonempty open subset of $$\mathbb{R}^d$$. Let $$K$$ be a compact subset of $$\Omega$$. Now take $$V=\mathcal{D}_{K,\Omega}$$, the space of $$C^{\infty}$$ functions $$\Omega\rightarrow\mathbb{R}$$ with support contained in $$K$$. Take $$A=\{||\cdot||_N\ |\ N=1,2,3\ldots\}$$ as in the question. Then $$\mathscr{T}_A$$ gives $$\mathcal{D}_{K,\Omega}$$ a LCTVS structure.

Example 2: Now take instead $$V=\mathcal{D}(\Omega)$$. Let $$B\subset s(V)$$ be the set of all seminorms $$\rho$$ on $$\mathcal{D}(\Omega)$$, such that for all compact $$K\subset\Omega$$, $$\rho\circ \iota_{K,\Omega}:\mathcal{D}_{K,\Omega}\rightarrow\mathbb{R}$$ is a continuous map. Here $$\iota_{K,\Omega}$$ is the inclusion map of $$\mathcal{D}_{K,\Omega}$$ into $$\mathcal{D}(\Omega)$$. Now equip $$\mathcal{D}(\Omega)$$ with the topology $$\mathscr{T}_B$$. This is the standard topology of $$\mathcal{D}(\Omega)$$.

Example 3: Again take $$V=\mathcal{D}(\Omega)$$. Let $$\mathbb{N}=\{0,1,\ldots\}$$, and denote the set of multiindices by $$\mathbb{N}^d$$. A locally finite family $$\theta=(\theta_{\alpha})_{\alpha\in\mathbb{N}^d}$$ of continous functions $$\Omega\rightarrow \mathbb{R}$$ is one such that for all $$x\in\Omega$$ there is a neighborhood $$V\subset\Omega$$, such that $$V\cap {\rm Supp}\ \theta_{\alpha}=\varnothing$$ for all but finitely many $$\alpha$$'s. For $$f\in\mathcal{D}(\Omega)$$, let $$||f||_{\theta}=\sup_{\alpha\in\mathbb{N}^d}\sup_{x\in\Omega} |\theta_{\alpha}(x)D^{\alpha}f(x)|\ .$$ Let $$C$$ be the set of seminorms $$||\cdot||_{\theta}$$ where $$\theta$$ runs over all such locally finite families. Then $$\mathscr{T}_C$$ is also the standard topology of $$\mathcal{D}(\Omega)$$. Namely, $$\mathscr{T}_C=\mathscr{T}_B$$, where $$B$$ is the set of seminorms from the previous example [Exercise 4: prove this equality].

Remark 2: One can prove the above equality of topologies by showing that the identity map is a homeomorphism from $$\mathcal{D}(\Omega)$$ with the topology $$\mathscr{T}_B$$ to $$\mathcal{D}(\Omega)$$ with the topology $$\mathscr{T}_C$$, using the above criterion of continuity for multilinear maps (for $$n=1$$).

And for some more practice, Exercise 5: Prove that pointwise multiplication is continuous from $$\mathcal{D}(\Omega)\times \mathcal{D}(\Omega)$$ with the product topology, to $$\mathcal{D}(\Omega)$$. For the solution of the last exercise see: https://mathoverflow.net/questions/234025/why-is-multiplication-on-the-space-of-smooth-functions-with-compact-support-cont/234503#234503

Finally, details can be found in the lecture notes (rough draft) for a course on distributions I taught.

Answer for 2): Let $$N$$ be any positive integer. Since $$\phi$$ is non-negative it follows that $$\psi_m(N+\frac 1 2)\geq \frac 1 N \phi (N+\frac 1 2-N)=\frac 1 N \phi (\frac 12 )$$ whenever $$m \geq N$$. If $$\psi = \lim \psi_m$$ we get $$\psi (N+\frac 1 2) \geq \frac 1 N \phi (\frac 1 2) >0$$ for all $$N$$ . Hence $$\psi$$ does not have compact support.