A note on vector valued anisotropic Sobolev spaces \(L^ p_{\Gamma}(E)\) (Q1116418)

Let E be a Fréchet space and let \(\Gamma\) be a finite nonempty subset of \({\mathbb{N}}^ n\) such that if \((\alpha_ j)\in \Gamma\) then \((\beta_ j)\in \Gamma\) whenever \(0\leq \beta_ j\leq \alpha_ j\) for \(j=1,...,n\). In this note we prove that the vector-valued anisotropic Sobolev spaces \[ L^ p_{\Gamma}(E):=\{f\in L^ p(E):\quad D^{\alpha}f\in L^ p(E)\quad for\quad \alpha \in \Gamma \},\quad 1\leq p<\infty, \] have the approximation property if E has this property.