The Valdivia-Vogt structure table
presented in [3] contains the most prominent spaces of smooth functions occurring in the theory of
distributions together with their sequence-space representations.
Analogously its ``dual version''
contains the most prominent spaces of distributions together with their sequence space representations.
In [1] the existence of an isomorphism
\(\Phi\colon \mathcal{E}\rightarrow \mathbb{C}^{\mathbb{N}}\widehat{\otimes}s\)
such that every restriction to any other space in the structure table provides an
isomorphism between this space and its sequence space representation is shown.
We provide an explicit isomorphism between the spaces \(\mathcal{E}(\mathbb{R})=\mathcal{C}^{\infty}(\mathbb{R})\) and
\(\mathbb{C}^{\mathbb{N}}\widehat{\otimes}s\) as well as an explicit isomorphism between the spaces \(\mathcal{D}'(\mathbb{R})\) and
\(\mathbb{C}^{\mathbb{N}}\widehat{\otimes}s'\) which allow us to interpret the tables above as commutative diagrams. We use these
isomorphisms to construct a common basis for the spaces of smooth functions and one for the spaces of distributions in the tables
except \(\mathcal{D}_{L^\infty}\) and \(\mathcal{D}_{L^{\infty}}'\), those spaces being non-separable.
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References- C. Bargetz, Commutativity of the Valdivia-Vogt table of sequence space representations of spaces of smooth
functions , Math. Nachr. 287(1): 10-22, 2013.
- C. Bargetz, Explicit representations of spaces of smooth functions and distributions , Preprint, 2013.
- N. Ortner and P. Wagner, Explicit representations of L. Schwartz' spaces \(\mathcal{D}_{L^{p}}\) and
\(\mathcal{D}'_{L^p}\) by the sequence spaces \(s\widehat{\otimes}\ell^p\) and \(s'\widehat{\otimes}\ell^p\), respectively,
for \(1<p<\infty\), J. Math. Anal. Appl., 404(1): 1-10.
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