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Let $X$ be a real vector space and let $P$ be a family of seminorms on $X$. We say that $P$ is filtering if for any $p_1,p_2\in P$ we can find $q\in P$ and $c_1,c_2>0$ such that $c_1p_1\le q$ and $c_2p_2\le q$ both hold on $X$. I got no problem of constructing a filtering family of seminorms. I want to construct a family of seminorms which is not filtering. I tried but I can't. Can you please help me.

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up vote 3 down vote accepted

For example, $X=C(\mathbb R)$, the space of real-valued continuous functions on $\mathbb R$, and the seminorms $p_n(f)=\sup_{[n,n+1]}|f|$, $n\in\mathbb Z$. There is no seminorm in $P$ that majorizes both $p_1$ and $p_2$.

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