If I haven't committed any error in my proof, the space of continous applications mapping a compact space $X$ into $\mathbb{C}$ or $\mathbb{K}$ is complete with the metric defined by $d(f,g):=\sup_{x\in X}|f(x)-g(x)|=\max_{x\in X}|f(x)-g(x)|$.
Am I right?
Thank you very much for any answer!!!
For $\mathbb C$ it is OK, by showing that if $(f_n)_n$ is Cauchy then so is $(f_n(x))_n$ for each $x$.
For $\mathbb K$, it depends: if it is $\mathbb R$, the same argument as for $\mathbb C$ holds. But we have to be more careful when $\mathbb K$ is not a complete field.