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

Abstract

Let $K\subset\R^d$ 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 $A_k=\{(f_\alpha)_{\alpha\in\N_0^d}\in C(K)^{\N_0^d}\,:\, \|f\|_k:=\sup_{x\in K} \frac{|f^{(\alpha)}(x)|}{\alpha!} k^{-|\alpha|}< +\infty\}.$ Based on this description we give in the present note an explicit fundamental system of seminorms for $A(K)$.