Isomorphisms between spaces of test functions and sequence spaces
Check the following sources:
MR0688001 Reviewed Vogt, Dietmar Sequence space representations of spaces of test functions and distributions. Functional analysis, holomorphy, and approximation theory (Rio de Janeiro, 1979), pp. 405–443, Lecture Notes in Pure and Appl. Math., 83, Dekker, New York, 1983. (Reviewer: M. Valdivia)
M. Valdivia: Topics in locally convex spaces, North Holland, 1982.
A construction for the isomorphism not using the Pełczyński decomposition method can be found in
- C. Bargetz: Explicit representations of spaces of smooth functions and distributions. J. Math. Anal. Appl. 424: 149–1505. 2015 DOI: 10.1016/j.jmaa.2014.12.009
- C. Bargetz: Commutativity of the Valdivia–Vogt table of representations of function spaces. Mathematische Nachrichten 287(1): 10–22, 2014, DOI: 10.1002/mana.201200258.
I do not think that the basis constructed in the first of the above papers really has the properties you would like to have.
The construction of the isomorphism on the other hand consists of a series of relatively simple steps and it is completely constructive. More precisely, in the one dimensional case, the construction works as follows: First, using Seeley's extension operator, the function is "split" into a sequence of smooth functions on the unit interval $[0,1]$ which together with all derivatives vanish at the right boundary point $1$. Then, using an explicit formula each of these smooth functions is mapped to an element of $s$.
Some topological properties of $s^{(\mathbb{N})}$ and related spaces are dicussed in
- C. Bargetz: Completing the Valdivia–Vogt tables of sequence-space representations of spaces of smooth functions and distributions. Monatsh. Math. 177(1): 1–14, 2015. DOI: 10.1007/s00605-014-0650-2