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I am studying the properties of function spaces in the context of distribution theory and Fourier analysis, and I would like to understand the mathematical motivations behind the choice of specific topologies on these spaces.

Test Functions: The space of test functions, $\mathcal{D}(\Omega)$, is typically equipped with the inductive limit topology. What is the rationale for this choice? Why do we not prefer the topology generated by seminorms on this space instead?

Schwartz Space: Conversely, the Schwartz space, $\mathcal{S}(\mathbb{R}^n)$, uses a topology generated by seminorms. What advantages does this topology provide? Why is the inductive limit topology not suitable for the Schwartz space?

I am looking for a deeper mathematical motivation that explains why these topologies are preferred over others, considering their implications for the convergence of sequences, the behavior of functions at infinity, and their applications in analysis. Any insights or references would be greatly appreciated!

P.S.: Is it to get non empty dual? In other words, is the dual of $D(\Omega)$ empty set with respect to the topology generates by the seminorm topology and vice versa?

Daniele Tampieri
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Celestina
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    The definition of $\varphi\in \mathscr D(\Omega)$ starts with an equistence quantifier (there is a compact set....) wich naturally yields an inductive limit. The definition of $f\in \mathscr S(\mathbb R^n)$ starts with a universal quantifier (for each order of differentiation and each polynomial...) which naturally yields a projective limit. Of course, one can describe the topology of $\mathscr D(\Omega)$ by seminorms, a rather concrete system is used in Hörmander's The analysis of Partial Differential Operators I. – Jochen Oct 10 '24 at 15:27
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    As Jochen said one can describe the topology of $\mathscr{D}$ using explicit seminorms. The best that I know are the ones due to Garding, Lions and Horvath and explained in example 3 of https://math.stackexchange.com/questions/3510982/doubt-in-understanding-space-mathscr-d-omega/3511753#3511753 – Abdelmalek Abdesselam Oct 15 '24 at 21:43

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