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Definition. A family $P$ of semi-norms on a vector space $X$ is called directed if for any $p_1,p_2\in P$ there exist $p\in P$ and $C>0$ such that $p_1(x)+p_2(x)\leq Cp(x)$ for all $x\in X$.

Let $P$ be a family of semi-norms that generates the topology of a locally convex space $X$. Denote by $C([a,b],X)$ the space of continuous functions from a compact interval $[a,b]$ into $X$. For each $p\in P$, define $$q_p(f)=\sup_{t\in[a,b]}p(f(t))$$ for all $f\in C([a,b],X)$. Then the family $\Gamma$ given by $$\Gamma=\{q_p:p\in P\}$$ is a family of semi-norms on the space $C([a,b],X)$.

Question: If $P$ is directed, does it follow that $\Gamma$ is also directed?

Juniven Acapulco
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    Yes. Let $q_1,q_2\in \Gamma$. Let $p_i\in P$ with $q_i=q_{p_i}$. Choose $p\in P$ and $C\in\mathbb{R}$ such that $p_1+p_2\leq Cp$. In particular, $p_i\leq Cp$, and it follows that $q_i\leq Cq_p$, so $q_1+q_2\leq 2Cq_p$. –  May 25 '15 at 01:04
  • @ Questioner, Thank you very much. – Juniven Acapulco May 25 '15 at 18:15

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