I feel a bit uneasy about the proof of the following Theorem in Rudin's Functional Analysis, 2nd edition, p. 152-153. It says that for the space $\mathcal{D}(\Omega)$ and a certain systems $ \beta, \tau$ of its subsets, the following Theorem is true:

  1. $\tau$ is a topology in $\mathcal{D}(\Omega)$ and $\beta$ is a local base for this $\tau.$
  2. $\tau$ makes $\mathcal{D}(\Omega)$ into locally convex topological vector space.

The context is the following: the system $\tau$ is defined in the following rather convoluted way by means of the system $\beta$ (the overall construction is known as a strict inductive limit topology). We suppose that $\Omega$ is a fixed open set in $\mathbb{R}^{n}$ (which, technically, could be also the $\mathbb{R}^n$ itself) and we first define, for another fixed compact set $K,$ a subspace of $C^{\infty}(\Omega)$ functions as the set $$ \mathcal{D}_K(\Omega) = \{\phi \in C^{\infty}(\Omega): \mathrm{supp}\ \phi \subset K \}. $$ As a next step, we define a topology $\tau_K$ on $\mathcal{D}_K(\Omega)$ as a topology defined by the countable system of seminorms $\|\phi\|_{N} = \max \{|D^{\alpha}\phi(x)|: x \in \Omega, |\alpha| \leq N \}$ (here $N$ runs through $\mathbb{N} \cup \{0\}$). The general considerations from Chapter 1 of the book then show that the system $\tau_K$ is indeed a topology with a local base of neighbourhoods of zero given by the sets $$ V_{N} = \{\phi \in \mathcal{D}_K(\Omega): \|\phi\|_{N} < 1/N \}, $$ $N \geq 1,$ which is in fact metrizable by a translation invariant metric $d$ in such a way that the space $(\mathcal{D}_K(\Omega), d)$ is complete as a metric space. Having a topology $\tau_K$ on $\mathcal{D}_K(\Omega)$ for every $K,$ a compact subset of $\Omega,$ we can define the space $$ \mathcal{D}(\Omega) = \bigcup_{K \subset \Omega} \mathcal{D}_K(\Omega), $$ where $K$ runs through all compact subsets of $\Omega,$ and a system $\beta \subset \tau$ as the collection of all convex balanced sets $W \subset \mathcal{D}(\Omega)$ such that $\mathcal{D}_K(\Omega) \cap W \in \tau_K$ for every compact $K \subset \Omega.$ Finally, $\tau$ is the collection of all possible unions of sets of the form $\phi + W,$ where $\phi \in \mathcal{D}(\Omega)$ and $W \in \beta.$

I have multiple problems with the proof of the theorem above:

  1. Inherent in the construction of the topology $\tau_K$ on $\mathcal{D}_K(\Omega)$ is the assumption that we can always construct a compact exhaustion of $\Omega.$ This can be done in several different ways. First question: from where does it follow that the topology $\tau_K$ is independent on the actual exhaustion?
  2. The proof actually does not show that the system $\beta$ is a local base for the topology $\tau,$ neither does it prove that the resulting space is a locally convex topological vector space. The latter obviously follows from the former, since the members of $\beta$ are all convex. So the second question: why is $\beta$ a local base of zero for $\tau.$
  3. When showing that together with $V_1, V_2 \in \tau,$ the intersection $V_1 \cap V_2$ lies in $\tau,$ the author picks a $\phi \in V_1 \cap V_2$ (this is without any loss of generality since for $V_1 \cap V_2 = \emptyset$ everything is clear) and he claims (as is obvious from the definitions, $\phi \in \phi_1 + W_1$ and $\phi \in \phi_2 + W_2,$ where $\phi_1,\phi_2 \in \mathcal{D}(\Omega)$ and $W_1,W_2 \in \beta$ and we can obviously choose $K$ such that the corresponding $\mathcal{D}_K(\Omega)$ contains both $\phi_2, \phi_2,$) that since $\mathcal{D}_K(\Omega) \cap W_i \in \tau_K,$ we have $$ \phi - \phi_i \in (1- \delta_i) W_i $$ for some $\delta_i > 0.$ My third question is: how does the purported existence of the $\delta_i$s follow from the fact that $\mathcal{D}_K(\Omega) \cap W_i \in \tau_K?$
  4. When showing that one-point sets are closed, the author says that the set $W = \{\phi \in \mathcal{D}(\Omega): \|\phi\|_0 < \|\phi_1 - \phi_2\|_0 \},$ where $\phi_1$ and $\phi_2$ are two distinct elements of $\mathcal{D}(\Omega),$ is in $\beta.$ My fourth question: Why?
  5. (soft) My overall impression about the book by Rudin is that it is written by an expert for (almost) experts. Is there any alternative to it that would contain the same portion of material but not written in such a harsh manner?

My attempts:

  1. The only thing that came to my mind is the following fact that I am able to show: any locally convex topology generated by a system of seminorms is the coarsest one, for which the seminorms are continuous. Is this somehow helpful here?
  2. To show that $\beta$ is a local base of zero for $\tau$ we have to prove that $\beta$ is a collection of $\tau-$neighbourhoods of zero such that for any $\tau-$negihbourhood $V \ni \mathbf{0}$ there is $U \in \beta$ such that $\mathbf{0} \in U \subset V.$ Since every $W \in \beta$ is balanced, it contains the $\mathbf{0}$ function. Furthermore $\mathbf{0} + W$ is obviously a member of $\tau$ for every $W \in \beta,$ being a union of just one set. These two facts together show that $\beta$ is indeed a system of (open) neighbourhoods of the zero function. To show the other part, we pick a $\tau-$neighbourhood $V \ni \mathbf{0}.$ If we could show that an arbitrary set $G$ is open in $\mathcal{D}(\Omega)$ iff for every $\phi \in G$ there is $F \in \beta$ such that $\phi + F \subset G,$ then in particular for $\mathbf{0},$ there would be a $W \in \beta$ such that $\mathbf{0} + W = W \subset V,$ which would indeed complete the proof. Now it is obvious that if for a set $G \subset \mathcal{D}(\Omega)$ we can find for every $\phi \in G$ such an $F \in \beta$ that $\phi + F \subset G,$ then $G$ is obviously open in $\mathcal{D}(\Omega).$ Is the converse also true?
  3. Knowing that $\mathcal{D}_K \cap W_i$ are open in $\mathcal{D}_K$ could mean that $W_i$ is a $\tau_K$ neighborhood of $\phi - \phi_i$ so that there should be a set $V_{N_i}$ of the form $$ V_{N_i} = \{\phi \in \mathcal{D}_K(\Omega): \|\phi\|_{N_i} < 1/(N_i) \}, $$ such that $\phi - \phi_i \in V_{N_i} \subset W_i,$ but this does not seem to be very helfpful.
  4. We would like to check that $\mathcal{D}_K(\Omega) \cap W \in \tau_K$ for arbitrary compact $K\subset \Omega.$ For this it would be enough to find to every $\phi \in \mathcal{D}_K(\Omega) \cap W$ such an $N$ that $\phi + V_N \subset \mathcal{D}_K(\Omega) \cap W.$ Correct? If so, what $N$ does the job and why?
  • $\begingroup$ It takes too much to write the whole answer, but together with Rudin's book, I advise you to consult also "A First Course in Sobelev Spaces" by Leoni. There is a good explanation of the topology on the test function space. $\endgroup$ – Andrew Apr 10 '18 at 22:01
  • $\begingroup$ @Andrew: you are right, the book that you refer to gives a much more detailed ellaboration of the ideas contained in Rudin, explaining most of my questions (namely 2,3 and 4),thank you. However, it seems that it does not address my question 1. $\endgroup$ – Jorge.Squared Apr 11 '18 at 7:16

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