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\begin{enumerate}
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\item For any $U \subset E$ convex and balanced, if $U$ absorbs every bounded set of $E$, then $U \in \cn_E(0)$.
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\item For any seminorm $\rho: E \to [0, \infty)$ that is bounded on all bounded sets of $E$, $\rho$ is continuous.
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\end\{enumerate\}
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\end{enumerate}
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If the above holds, then $E$ is a \textbf{bornological space}.
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