Problem 4 section 4.3 from Kreyszig Functional analysis book

Question

Can somebody please give a hint for this problem from Kreyszig book -

Let $p$ be defined on Vector Space $X$ and satisfies $p(x+y)\leq p(x) + p(y)$ and for every scalar $a$, $p(ax) = |a| p(x)$. Show that for any given $x'\in X$ there exists a linear functional $f'$ on $X$ such that $f'(x') = p(x')$ and $|f'(x)| \le p(x)$ for all $x \in X$.

Answer 1

HINT

Define $f'_0$ on the span of $x'$ by $f'_0(\alpha x')=\alpha p(x')$. Then extend using the Hahn-Banach theorem.