aa r X i v : . [ m a t h . F A ] J un A characteristic property of the space s dietmar vogt Abstract
It is shown that under certain stability conditions a complemented subspace ofthe space s of rapidly decreasing sequences is isomorphic to s and this conditioncharacterizes s . This result is used to show that for the classical Cantor set X the space C ∞ ( X ) of restrictions to X of C ∞ -functions on R is isomorphic to s , socompleting the theory developed in [7]. In the present note we study the space s of rapidly decreasing sequences, that is, thespace s = { x = ( x , x , . . . ) : lim n x n n k = 0 for all k ∈ N } . Equipped with the norms k x k k = sup n | x n | ( n + 1) k it is a nuclear Fr´echet space. It isisomorphic to many of the Fr´echet spaces which occur in analysis, in particular, spacesof C ∞ -functions.It is easily seen that instead of the sup-norms we might use the norms | x | k = (cid:0) X n | x n | ( n + 1) k (cid:1) / which makes s a Fr´echet-Hilbert space.More generally, we define for any sequence α : 0 ≤ α ≤ α ≤ր + ∞ the power seriesspace of infinite typeΛ ∞ ( α ) := { x = ( x , x , . . . ) : | x | t = ∞ X n =0 | x n | e tα n < ∞ for all t > } . Equipped with the hilbertian norms | · | k , k ∈ N , it is a Fr´echet-Hilbert space. It isnuclear if, and only if, lim sup n log n/α n < ∞ . With this definition s = Λ ∞ ( α ) with α n = log( n + 1). Key words and phrases: space s , stability condition, Cantor set k · k ≤ k · k ≤ . . . hasproperty (DN) if ∃ p ∀ k ∃ K, C > k · k k ≤ C k · k p k · k K . In this case k · k p is called a dominating norm. E has property (Ω) if ∀ p ∃ q ∀ m ∃ < θ < , C > k · k ∗ q ≤ C k · k ∗ pθ k · k ∗ m − θ . Here we set for any continuous seminorm k·k and y ∈ E ′ the dual, extended real valued,norm k y k ∗ = sup {| y ( x ) | : x ∈ E, k x k ≤ } .By Vogt-Wagner [8] a Fr´echet 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 Fr´echet spaces,going back to Mityagin, whether every complemented subspace of s has a basis. If ithas 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 methodgoing back to Terzio˘glu [4] which we describe now.Let X be a vector space and A ⊂ B absolutely convex subsets of X . We set δ n ( A, B ) := inf { δ > F ⊂ X, dim F ≤ n with A ⊂ δB + F } . 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 k · k p is a dominating norm and for p wechoose q > p according to property (Ω). We set α n = − log δ n ( U q , U p )where U k = { x ∈ E : k x k k ≤ } . The space Λ ∞ ( α ) is called the associated power seriesspace and E ∼ = Λ ∞ ( α ) if it has a basis.If lim sup n α n /α n < ∞ then, by Aytuna-Krone-Terzio˘glu [2, Theorem 2.2], E ∼ =Λ ∞ ( α ). This is, in particular, the case if E is stable, that is, if E ⊕ E ∼ = E .For all that and further results of structure theory of infinite type power series spacessee [6], for results and unexplained notation of general functional analysis see [3]. Lemma 2.1
Let E be a complemented subspace of s , k · k a dominating hilbertiannorm and k · k a hilbertian norm chosen for k · k according to (Ω) . If there is a linearisomorphism ψ : E ⊕ E → E such that k x k + k y k ≤ C k ψ ( x ⊕ y ) k k ψ ( x ⊕ y ) k ≤ C ( k x k + k y k ) then E ∼ = s . roof. For x ⊕ y ∈ E ⊕ E we set ||| ( x, y ) ||| := ( k x k + k y k ) / and ||| ( x, y ) ||| :=( k x k + k y k ) / . With new constants C k we have ||| x ⊕ y ||| ≤ C k ψ ( x ⊕ y ) k and k ψ ( x ⊕ y ) k ≤ C ||| x ⊕ y ||| . (1)To calculate the associated power series space for E we set: α n = − log δ n ( U , U ) where U k = { x ∈ E : k x k k ≤ } ,β n = − log δ n ( V , V ) where V k = { x ⊕ y ∈ E ⊕ E : ||| x ⊕ y ||| k ≤ } . Due to the estimates (1) we have1 C ψ ( V ) ⊂ U ⊂ U ⊂ C ψ ( V )and therefore δ n ( V , V ) = δ n ( ψV , ψV ) ≤ C C δ n ( U , U )which implies α n ≤ β n + d with d = log C C .By explicit calculation of the Schmidt expansion of the canonical map j between thelocal Hilbert spaces of ||| · ||| and ||| · ||| and by use of the fact that singular numbersand Kolmogoroff diameters coincide, we obtain that β n = β n +1 = α n for all n ∈ N .Therefore we have α n ≤ β n + d = α n + d for all n ∈ N and this implies α k ≤ α + k d for all k ∈ N . For n ∈ N we find k ∈ N such that 2 k − ≤ n ≤ k and we obtain α n ≤ α k ≤ α + k d ≤ ( α + d ) + d log n .Since E ⊂ s , which implies the left inequality below, we have shown that there is aconstant D > D log n ≤ α n ≤ D log n for large n ∈ N . This implies that Λ ∞ ( α ) = s . ✷ A Fr´echet-Hilbert space E is called normwise stable if it admits a fundamental systemof hilbertian seminorms for which there is an isomorphism ψ : E ⊕ E → E such that1 C k ( k x k k + k y k k ) ≤ k ψ ( x ⊕ y ) k k ≤ C k ( k x k k + k y k k )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 heorem 2.3
Let the Fr´echet-Hilbert space E be a complemented subspace of s , k · k a dominating norm and k · k be a norm chosen according to (Ω) . Let P be a linearprojection in E , continuous with respect to k · k . We set E = R ( P ) , E = N ( P ) and assume that there are linear isomorphisms ψ j : E → E j , j = 1 , , continuous withrespect to k · k such that ψ − is continuous with respect to k · k . Then E ∼ = s . Proof.
We set ψ ( x ⊕ y ) := ψ ( x ) + ψ ( y ) and obtain with suitable constants: k x k + k y k ≤ C ′ ( k ψ ( x ) k + k ψ ( y ) k ) ≤ C k ψ ( x ) + ψ ( y ) k = C k ψ ( x ⊕ y ) k k ψ ( x ⊕ y ) k = k ψ ( x ) + ψ ( y ) k ≤ k ψ ( x ) k + k ψ ( y ) k ≤ C ( k x k + k y k ) . Lemma 2.1 yields the result. ✷ An interesting application of this result is the following. Let X ⊂ [0 ,
1] be the classicalCantor set and C ∞ ( X ) := { f | X : f ∈ C ∞ [0 , } = { f | E : f ∈ C ∞ ( R ) } . The space C ∞ ( X ) equipped with the quotient topology is a nuclear Fr´echet space and, since C ∞ [0 , ∼ = s isomorphic to a quotient of s , hence has property (Ω). By a theoremof Tidten [5] it has also property (DN). Therefore it is isomorphic to a complementedsubspace 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 C ∞ ( X ) ∼ = C ∞ ( X ∩ [0 , / ⊕ C ∞ ( X ∩ [2 / , ∼ = C ∞ ( X ) ⊕ C ∞ ( X )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 thediametral dimensions of E ( X ) and s coincide, from where, by means of the Aytuna-Krone-Terzio˘glu Theorem, on can derive the same result.Referring to the terminology of [7] we have also shown that A ∞ ( X ) ∼ = s which completesthe theory developed in [7]. References [1] Arslan, B., Goncharov, A. P., Kocatepe, Spaces of Whitney functions on Cantor-type sets,Canad. J. Math. (2002), 225–238.[2] Aytuna A., Krone J., Terzio˘glu T., Complemented infinite type power series subspaces ofnuclear Fr´echet spaces, Math. Ann. (1989), 193–202.
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Introduction to Functional Analysis , Clarendon Press, Oxford.[4] Terzio˘glu T., On the diametral dimension of some classes of F-spaces, J. Karadeniz Uni.Ser. Math.-Phys. (1985), 1–13.[5] M. Tidten: Fortsetzungen von C ∞ -Funktionen, welche auf einer abgeschlossenen Mengein R n definiert sind, Manuscripta Math. (1979), 291–312.[6] Vogt D.,Structure theory of power series spaces of infinite type, Rev. R. Acad. Cien. SerieA. Mat. 97 (2), (2003), 1–25[7] Vogt, D., Restriction spaces of A ∞ , to appear in Rev. Mat. Iberoamericana 29.4 (2013)[8] Vogt D., Wagner M. J., Charakterisierung der Quotientenr¨aume von s und eine Vermutungvon Martineau, Studia Math. (1980), 225–240.(1980), 225–240.