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I have been reading about inductive limit topology from the book "A Course in Functional Analysis" by J B Conway. My motivation for reading this was to understand Fourier Transforms and Distribution Theory given in the book "Integral Geometry and Radon Transforms" by S. Helgason.

For reference to this question, I will be using the following definition of inductive limit topology:

Definition (Inductive system): A pair $(X_i, \left\lbrace X_i : i \in I \right\rbrace)$ is an inductive system if $X$ is a vector space, each $X_i$ is a linear manifold (vector subspace) of $X$, each $X_i$ has a topology $\mathscr{T}_i$ which is locally convex such that

  1. $I$ is a directed set and whenever $i \leq j$ (with $i, j \in I$), we have $X_i \subseteq X_j$.
  2. Whenever $i \leq j$ and $U_j \in \mathscr{T}_j$, we have $U_j \cap X_i \in \mathscr{T}_i$.
  3. $X = \bigcup\limits_{i \in I} X_i$.

The book by J B Conway gives the topology on $D \left( \mathbb{R}^n \right)$, the set of all compactly supported smooth functions, by defining for each compact set $K \subseteq \mathbb{R}^n$, the family of seminorms on $D_K \left( \mathbb{R}^n \right) := \left\lbrace \phi \in D \left( \mathbb{R}^n \right) | \text{supp } \phi \subseteq K \right\rbrace$ as

$$p_{k, m} \left( \phi \right) = \sup \left\lbrace \left| \phi^{\left( \alpha \right)} \left( x \right) \right| : \left| \alpha \right| \leq m, x \in K \right\rbrace.$$

Here, $\alpha = \left\lbrace \alpha_1, \alpha_2, \cdots, \alpha_n \right\rbrace \in \left( \mathbb{N} \cup \left\lbrace 0 \right\rbrace \right)^n$, and $\left| \alpha \right| = \sum\limits_{i = 1}^{n} \alpha_i$. Also, we have the notation

$$\phi^{\left( \alpha \right)} = \dfrac{\partial^{\left| \alpha \right|} \phi}{\partial^{\alpha_1} x_1 \partial^{\alpha_2} x_2 \cdots \partial^{\alpha_n} x_n}.$$

I have verified that this indeed gives an inductive limit topology on $D \left( \mathbb{R}^n \right)$.

However, in the book by Helgason, the author defines the family of seminorms as

$$p_{K, m} \left( \phi \right) = \sum\limits_{\left| \alpha \right| \leq m} \sup \left\lbrace \left| \phi^{\alpha} \left( x \right) \right| : x \in K \right\rbrace.$$

My question is that are these two topologies different? Or are they equivalent? I believe that they are equivalent, but I am not quite getting the idea to prove it.

Abdelmalek Abdesselam
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Aniruddha Deshmukh
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  • Two questions: in the definition of $p_{K,m}$, are you missing an absolute value? Have you already seen that both spaces are complete? –  May 05 '21 at 11:43
  • @Caffeine: I have edited the aboslute value. As for the second question, no, I have not seen that they are both complete. How would that help? – Aniruddha Deshmukh May 05 '21 at 11:59
  • Once you have proved that they are complete, you can easily prove that they are Frechet spaces. From that you can use the usual machinery (i.e. closed graph theorem and open map) to prove that the identity (from one topology to the other) is a homeomorphism. –  May 05 '21 at 12:18
  • @AniruddhaDeshmukh: Please see https://math.stackexchange.com/questions/3510982/doubt-in-understanding-space-d-omega/3511753#3511753 – Abdelmalek Abdesselam May 06 '21 at 15:45

1 Answers1

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You have an inequality like : $$p_{K,m}^{\rm{Conway}} \leqslant p_{K,m}^{\rm{Helgason}}\leqslant C_{n,m} p_{K,m}^{\rm{Conway}}$$

where $C_{n,m}$ is the number of multi-indices with $|\alpha|\leq m$.

This should be enough to show directly that the open sets of both topologies are the same

SolubleFish
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