A characteristic property of the space s

It is shown that under certain stability conditions a complemented subspace of the space $s$ of rapidly decreasing sequences is isomorphic to $s$ and this condition characterizes $s$. This result is used to show that for the classical Cantor set $X$ the space $C_\infty(X)$ of restrictions to $X$ of $C^\infty$-functions on $\R$ is isomorphic to $s$, so completing the theory developed in"Restriction spaces of $A^\infty$", to appear in Rev. Mat. Iberoamericana 29.4 (2013)


Introduction
In the present note we study the space s of rapidly decreasing sequences, that is, the space s = {x = (x 0 , x 1 , . . . ) : lim n x n n k = 0 for all k ∈ N}.
Equipped with the norms x k = sup n |x n |(n + 1) k it is a nuclear Fréchet space. It is isomorphic to many of the Fréchet spaces which occur in analysis, in particular, spaces of C ∞ -functions.
It is easily seen that instead of the sup-norms we might use the norms |x| k = n |x n | 2 (n + 1) 2k 1/2 which makes s a Fréchet-Hilbert space.
By Vogt-Wagner [8] a Fréchet space E is isomorphic to a complemented subspace of s if, and only if, it is nuclear and had properties (DN) and (Ω).
It is a long standing unsolved problem of the structure theory of nuclear Fréchet spaces, going back to Mityagin, whether every complemented subspace of s has a basis. If it has a basis then it is isomorphic to some power series space Λ ∞ (α). The space Λ ∞ (α) to which it is isomorphic, if it has a basis, can be calculated in advance by a method going back to Terzioglu [4] which we describe now.
Let X be a vector space and A ⊂ B absolutely convex subsets of X. We set It is called the n-th Kolmogoroff diameter of A with respect to B.
If now E is a complemented subspace of s, that is, E is nuclear and has properties (DN) and (Ω), then we choose p such that · p is a dominating norm and for p we choose q > p according to property (Ω). We set is called the associated power series space and E ∼ = Λ ∞ (α) if it has a basis.
For all that and further results of structure theory of infinite type power series spaces see [6], for results and unexplained notation of general functional analysis see [3].

Main result
Lemma 2.1 Let E be a complemented subspace of s, · 0 a dominating hilbertian norm and · 1 a hilbertian norm chosen for · 0 according to (Ω). If there is a linear isomorphism ψ : E ⊕ E → E such that Proof. For x ⊕ y ∈ E ⊕ E we set |||(x, y)||| 0 := ( x 2 0 + y 2 0 ) 1/2 and |||(x, y)||| 1 := ( x 2 1 + y 2 1 ) 1/2 . With new constants C k we have To calculate the associated power series space for E we set: Due to the estimates (1) we have By explicit calculation of the Schmidt expansion of the canonical map j 0 1 between the local Hilbert spaces of ||| · ||| 1 and ||| · ||| 0 and by use of the fact that singular numbers and Kolmogoroff diameters coincide, we obtain that β 2n = β 2n+1 = α n for all n ∈ N 0 . Therefore we have α 2n ≤ β 2n + d = α n + d for all n ∈ N 0 and this implies α 2 k ≤ α 1 + k d for all k ∈ N 0 . For n ∈ N we find k ∈ N such that 2 k−1 ≤ n ≤ 2 k and we obtain α n ≤ α 2 k ≤ α 1 + k d ≤ (α 1 + d) + d log n.
Since E ⊂ s, which implies the left inequality below, we have shown that there is a constant D > 0 such that 1 D log n ≤ α n ≤ D log n for large n ∈ N. This implies that Λ ∞ (α) = s. ✷ A Fréchet-Hilbert space E is called normwise stable if it admits a fundamental system of hilbertian seminorms for which there is an isomorphism ψ : for all k. Since, clearly, s is normwise stable we have shown.
Theorem 2.2 E ∼ = s if, and only if, E is isomorphic to a complemented subspace of s and normwise stable.
We may express Lemma 2.1 also in the following way: 3 Let the Fréchet-Hilbert space E be a complemented subspace of s, · 0 a dominating norm and · 1 be a norm chosen according to (Ω). Let P be a linear projection in E, continuous with respect to · 0 . We set E 1 = R(P ), E 2 = N (P ) and assume that there are linear isomorphisms ψ j : E → E j , j = 1, 2, continuous with respect to · 1 such that ψ −1 is continuous with respect to · 0 . Then E ∼ = s.

Application
An interesting application of this result is the following. Let X ⊂ [0, 1] be the classical Cantor set and C ∞ (X) : The space C ∞ (X) equipped with the quotient topology is a nuclear Fréchet space and, since C ∞ [0, 1] ∼ = s isomorphic to a quotient of s, hence has property (Ω). By a theorem of Tidten [5] it has also property (DN). Therefore it is isomorphic to a complemented subspace of s (see [8]).
We should remark that, due to the fact that X is perfect, we have C ∞ (X) = E(X) where E(X) denotes the space of Whitney jets on X, for which Tidten's result is formulated.
By obvious identifications we have and it is easily seen that this establishes normwise stability. Therefore we have shown Theorem 3.1 If X is the classical Cantor set, then C ∞ (X) ∼ = s.
It should be remarked that in [1] it has been shown that for the Cantor set X the diametral dimensions of E(X) and s coincide, from where, by means of the Aytuna-Krone-Terzioglu Theorem, on can derive the same result.
Referring to the terminology of [7] we have also shown that A ∞ (X) ∼ = s which completes the theory developed in [7].