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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

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