A Fundamental System of Seminorms for $A(K)$

Abstract

Let $K\subset {\mathbb{R}}^d$ be compact and $A(K)$ the space of germs of real analytic functions on $K$ with its natural (LF)-topology. This topology can be given by $A(K)=\limind_{k\to+\infty} A_k$ where $A_k=\{(f_{\alpha }{)}_{\alpha \in {\mathbb{N}}_0^d}\in C(K{)}^{{\mathbb{N}}_0^d}:\Vert f{\Vert }_k:={\sup }_{x\in K}\frac{|f^{(\alpha )}(x)|}{\alpha !}{k}^{-|\alpha |}<+\infty \}.$ Based on this description we give in the present note an explicit fundamental system of seminorms for $A(K)$.

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• Mathematical models for geometric control theory