Doubt in understanding Space $D(\Omega)$

I was reading Distribution form Rudin. I had 2 doubts in understanding space $$D(\Omega)$$

Doubts:

1)Why topology on $$D(\Omega)$$ and $$D_k$$ are same?

2) Why {$$\psi_m$$} is cauchy sequnce but its limit doesnot have compact support?

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 $D_K$). – reuns Jan 16 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. – GReyes Jan 16 at 8:18
• @GReyes Thanks a lot, Sir. Please can you suggest some books. I am interested in field of pde. – MathLover Jan 16 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. – Daniele Tampieri Jan 16 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"...) – paul garrett Feb 28 at 0:33

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.

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