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I am asking myself the following question:

Are space of test functions $\mathcal D$ and the space of distributions $\mathcal D'$ normed spaces (or even Banach spaces)?

My thought. I think that the answer is YES since I intuitively see that it is possible to define the norms in $\mathcal D'$. For examples, let $T\in \mathcal D'$, we may define the norm for $T$ as follows

$$||T||_{\mathcal D'}= \sup_{f\in \mathcal D: ||f||_{\infty} \leq 1} |T(f)|$$

or

$$||T||_{\mathcal D'}= \sup_{f\in \mathcal D: ||f||_{L^2} \leq 1} |T(f)|.$$

However, when I google I see that many papers try to put $\mathcal D'$ and $\mathcal D$ in some bigger Banach spaces than themselves, see e.g. this paper.

So I am really confusing. Am I right?

Leonard Neon
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    These are norms but they don't have anything to do with the standard topology on $\mathcal D'$. – Kavi Rama Murthy Nov 03 '21 at 23:31
  • @KaviRamaMurthy do you mean the norms $||.||\infty$ and $||.||{L^2}$ are not norms associating with topology of $\mathcal D$, then the norm $||.||_{\mathcal D'}$ defined like above is not norm of $\mathcal D'$ w.r.t. the standard topology of $\mathcal D'$. So is there available norm corresponding to the topology of $\mathcal D$? – Leonard Neon Nov 03 '21 at 23:39
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    No, there is no such norm. BTW every vector space has a norm so producing some norm is of no use. – Kavi Rama Murthy Nov 03 '21 at 23:50
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    The answer is No. To learn the topology of $\mathcal{D}$ see https://math.stackexchange.com/questions/3510982/doubt-in-understanding-space-d-omega/3511753#3511753 – Abdelmalek Abdesselam Nov 04 '21 at 19:26
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    Also, I don't see the benefit of the imbedding into larger Banach spaces. The spaces $\mathcal{D}$, $\mathcal{D}'$ with standard topologies are much better than any Banach space one can find on the market. – Abdelmalek Abdesselam Nov 04 '21 at 19:28
  • @AbdelmalekAbdesselam yeah thank you I will look at it, thank you so much! – Leonard Neon Nov 05 '21 at 17:06
  • The last two comments are not quite true. The paper quoted above was designed to provide the correct Hilbert space for the Feynman formulation of quantum mechanics. It contains the test functions as a continuous dense embedding, so the topologes are compatible.In addition this space was used to construct Feynman's time-ordered operator calculus and to prove the last two conjectures of Dyson on the foundations for QED. see Gill and Zachary "" Functional Analysis and the Feynman Operator Calculus, Springer, 2016. – Tepper L Gill Mar 06 '25 at 20:18

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