𝐴-differentiability and 𝐴-analyticity

Abstract

Let A A be a finite-dimensional commutative algebra over R \mathbb {R} and let C A r ( U ) C_{A}^{r}(U) , C ω ( U , A ) C^{\omega }(U,A) and O A ( U ) \mathcal { O}_{A}(U) be the ring of A A -differentiable functions of class C r , 0 ≤ r ≤ ∞ C^{r},\,0 \leq r \leq \infty , the ring of real analytic mappings with values in A A and the ring of A A -analytic functions, respectively, defined on an open subset U U of A n A^{n} . We prove two basic results concerning A A -differentiability and A A -analyticity: 1 s t 1^{st} ) O A ( U ) = C A ∞ ( U ) ⋂ C ω ( U , A ) \mathcal { O}_{A}(U) = C^{\infty }_{A}(U) \bigcap C^{\omega }(U,A) , 2 n d 2^{nd} ) O A ( U ) = C A ∞ ( U ) \mathcal { O}_{A}(U) = C^{\infty }_{A}(U) if and only if A A is defined over C \mathbb {C} .