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Suppose that we have a Hausdorff locally convex space with its topology $\tau$ and let $P(X)$ be a separating family of $\tau$-continuous semi-norms so that $\tau$ is generated by $P(X)$. How do we prove the following:

If $p_1,\cdots,p_n\in P(X)$, then we can find $p\in P(X)$ and a constant $k\ge 1$ such that for each $i\in \{1,\cdots,n\}$ and each $x\in X$, we have $p_i(x)\le k\cdot p(x)$.

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    Are there other assumptions about $P(X)$? for example, if $X$ is the space of continuous functions on $[0,1]$ and $\rho_x(f):=|f(x)|$, it gives a separating family of semi-norms.2012-11-09

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