A splitting theorem for the space of smooth functions. (English) Zbl 0912.46026
Consider an exact complex
\[
0\longrightarrow F \longrightarrow C^\infty \mathop{\longrightarrow}^{T_0}C^\infty \mathop{\longrightarrow}^{T_1}C^\infty \mathop{\longrightarrow}^{T_2}\cdots
\]
where \(T_n\) are (matrices of) linear differential operators with constant coefficients or convolution operators. It is proved that this complex splits from \(T_1\) on, i.e. \(T_n\) has for \(n\geq 1\) a continuous linear right inverse. The complexes which split completely are characterized.
The results are achieved by considering \(C^\infty\) as a graded Fréchet space, i.e. as a Fréchet space with a fixed associated countable projective system of Fréchet spaces. Abstract splitting results are proved for complexes of graded spaces. Conditions are given under which this abstract splitting is equivalent to the splitting of the corresponding nongraded complex. The approach is applicable for a larger class of spaces and operators.
The results are achieved by considering \(C^\infty\) as a graded Fréchet space, i.e. as a Fréchet space with a fixed associated countable projective system of Fréchet spaces. Abstract splitting results are proved for complexes of graded spaces. Conditions are given under which this abstract splitting is equivalent to the splitting of the corresponding nongraded complex. The approach is applicable for a larger class of spaces and operators.
Reviewer: Martin Väth (Würzburg)
MSC:
| 46E10 | Topological linear spaces of continuous, differentiable or analytic functions |
| 46A04 | Locally convex Fréchet spaces and (DF)-spaces |
| 35N15 | \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs |
| 46A22 | Theorems of Hahn-Banach type; extension and lifting of functionals and operators |
| 46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |
| 46A45 | Sequence spaces (including Köthe sequence spaces) |
| 35N05 | Overdetermined systems of PDEs with constant coefficients |
| 46A13 | Spaces defined by inductive or projective limits (LB, LF, etc.) |
Keywords:
splitting of exact sequence; linear differential operators with constant coefficients; graded Fréchet space; reduced spectrum; space of smooth functions; exact complex; convolution operators; continuous linear right inverseReferences:
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