# Doubt in section 1.46 Rudin Functional analysis

I was reading section 1.46 form Rudin Functional analysis

I had following doubt 1) Why linear functional $$\phi_x(f)=f(x)$$ is continous ?

2)Why $$D_k$$ is closed in $$C^\infty$$? (I do not understand the intersection of null space argument

3) why $$C^\infty$$ is complete?I know that over compact set it is complete using uniform convergence but i do not understand reasioning as set is open?

If $$x\in K_n$$ (every $$x\in\Omega$$ lies in some $$K_n$$), the continuity of $$\phi_x$$ follows from the fact that $$|\phi_x(f)|\leq p_n(f)$$ (considering the multi-index $$\alpha=(0,...,0)$$).
Then, $$D_K=\cap_{x\in K^c} \ker(\phi_x),$$ and as we've argued, $$\phi_x$$ is continuous for every $$x$$, so $$\ker(\phi_x)$$ is closed. The interesction of closed sets is closed.
For the completeness of $$C^{\infty},$$ Rudin has all the details in the text above. I'd recommend reading it again.