Proof of : $\cap_{f\in E'}Ker(f)= \left\{0 \right\}$ Let $(E,N)$ be a normed vector space.
The dimension of $E$ could be infinite.
Let $E'= \left\{f:E \rightarrow \mathbb{K} ,\quad f \quad linear \quad and \quad continuous \right\}$.
do we have $\cap_{f\in E'}Ker(f)=  \left\{0 \right\}$ ?

If $\mathbb K = \mathbb R$ or $\mathbb C$, then this is True by Hahn-Banach Theorem. 
I am not certain but you have more general version of Hahn-Banach for other fields such as p-adic fields. 
