Consequences of Hahn Banach Theorem

Question

Let $V$ be a normed vector space over $\mathbb{C}$

Let $U \subset V$ be a subspace of $V$

Let $p : V \to \mathbb{R}$ be a seminorm

Let $f : U \to \mathbb{C}$ be a continuous linear functional such that $\forall u \in U: |f(u)| \leq p(u)$

I would like to know if is it true that exist a continuous linear functional $F : V \to \mathbb{C}$ such that $$ \forall u \in U: f(u) = F(u) $$ $$ \forall v \in V: |F(v)| \leq p(v) $$ $$ ||F||=||f|| $$ Thanks.

Answer 1

COMMENT.-See at https://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem where the first "important consequence" is what you need and where you have a bibliography in English. In many books this consequence is the theorem itself. For example in the attached figure (a french source; the proof is not easy at all for a beginner).