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Lars Hormander, The Analysis of Linear Partial Differential Operators I, chapter II:

Let $X$ be an open set in $\mathbb {R}^n$. A distribution $u$ in $X$ is a linear form of $C_0^\infty(X)$ such that for every compact $K \subset X$ there exist constants $C$ and $k$ such that $$ |u(\phi)| \leq C \sum_{|\alpha| \leq k } {\rm sup} |\partial ^{\alpha} \phi|, \hspace{1cm} \forall \phi \in C_0^\infty (K) $$

My question is. How does this definition even come to mind? Is there a reason to define a distribution like that?

Sebastiano
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ric.san
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    One of the important things we want to do with distributions is differentiate them. It turns out that you therefore want $\phi \mapsto \partial^\alpha \phi$ (operator on test functions) to be continuous for all $\alpha$. L. Schwartz chose what you see here so that we get a complete locally convex topology on $C_0^\infty(X)$. – GEdgar Jan 12 '21 at 11:11
  • You haven't stated the definition correctly. Your version makes little sense: you say "for every compact $K$" and then don't mention $K$. – David C. Ullrich Jan 12 '21 at 12:12
  • @DavidC.Ullrich thank you for noticing. Now I think it's correct – ric.san Jan 12 '21 at 12:16
  • Since $K$ is not open you might want to clarify what $C^\infty_0(K)$ is... – David C. Ullrich Jan 12 '21 at 12:18
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    it's not specified in the book.. I think $K \supseteq {\rm supp} \ \phi$ – ric.san Jan 12 '21 at 12:26

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A locally convex space $X$ is a linear space with a topology generated by a family of seminorms. A seminorm is a function $p:X\rightarrow [0,\infty)$ with $p(x+y)\leq p(x)+p(y)$ but where $p(x) = 0$ does not necessarily imply that $x = 0$.

Now for such spaces, a linear functional $l:X\rightarrow \mathbb{C}$ is continuous if and only if

$|l(x)|\leq C\sum_{k\in 1}^{n}p_k(x)$

for some finite collection of seminorms which generate the topology. See for instance Theorem IV.3.1 in John B. Conways book on functional analysis.

The principle is the same as for normed spaces (which is a locally convex space generated by one seminorm), where the continuous linear functionals are those that satisfy

$$|l(x)|\leq C|x|$$ for all $x$.

Distributions are simply the continuous linear functionals acting on the space of test functions, with topology generated by the family of seminorms $\{p_\alpha\}$ where

$$p_{\alpha}(\phi) = \sup |\partial^\alpha \phi|.$$

For more on this one can have a look at Chapter IV in the book by John B. Conway

OgvRubin
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    Correct, up until when you say the $p_{\alpha}$ define the topology of the space of test functions. The topology is more complicated than that, see: https://math.stackexchange.com/questions/3510982/doubt-in-understanding-space-d-omega/3511753#3511753 – Abdelmalek Abdesselam Jan 12 '21 at 18:29