Splitting of exact sequences of Fréchet spaces in the absence of continuous norms
This paper studies the splitting of short exact sequences \[ 0 \rightarrow G \rightarrow E \rightarrow F \rightarrow 0 \tag{1} \] of nuclear Fréchet spaces the in the case that \(F\) does not necessarily have a continuous norm. The most important case is when \(E\) and \(F\) are spaces of \(C^{\infty}\) functions on an open subset \(\Omega\) of \(\mathbb{R}^n\). Valdivia and Vogt proved independently that this space is isomorphic to a countable product of copies of the Fréchet space \(s\) of rapidly decreasing sequences. The splitting theorem of Vogt and Wagner ensures that, if \(G\) has property \((\Omega)\) and \(F\) has property (DN), then the exact sequence (1) splits, i.e., the quotient map \(A:E \rightarrow F\) has a continuous linear right inverse. However, property (DN) implies the existence of a continuous norm. \textit{P. Domański} and \textit{D. Vogt} [J. Funct. Anal. 153, 203--248 (1998; Zbl 0912.46026)] developed a theory of graded Fréchet spaces and graded exact sequences, with important applications to the splitting of differential complexes. In the present paper, the author takes a different approach: instead of imposing conditions on the type of exactness, he assumes restrictions on the maps admitted in the exact sequence. A map is called seminorm-kernel continuous (SK-continuous) if it is continuous for the additive group topologies on \(E\) and \(F\) whose neighbourhoods of the origin are the kernels of the continuous seminorms. The condition that the quotient map \(A:E \rightarrow F\) is an SK-homomorphism is natural, since it is satisfied if the sequence (1) splits. Two new topological invariants \((DN_{loc})\) and \((\Omega_{\text{loc}})\) are introduced and studied. The main result of the article asserts that if the quotient map \(A:E \rightarrow F\) is an SK-homomorphism, \(F\) has property \((DN_{\text{loc}})\) and \(G\) has property \((\Omega_{\text{loc}})\), then the exact sequence (1) splits. This is applied to obtain a structure theory of subspaces, quotients and complemented subspaces of the countable product of the space \(s\) of rapidly decreasing sequences. The relation of the present point of view with the graded theory of Domański and Vogt is explained. Applications to the splitting of differential complexes are included. In particular, a characterization of the splitting of the whole complex is given in the case of \(C^{\infty}\)-functions, when the first operator has constant coefficients, which resembles the Phragmén-Lindelöf conditions of the work of Meise, Taylor and Vogt.
scientific article; zbMATH DE number 2107222
scientific article