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\begin{lemma}
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\label{lemma:continuous-seminorm}
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Let $E$ be a TVS over $K \in \RC$ and $[\cdot]: E \times E \to [0, \infty)$ be a seminorm on $E$, then the following are equivalent:
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Let $E$ be a TVS over $K \in \RC$ and $[\cdot]: E \to [0, \infty)$ be a seminorm on $E$, then the following are equivalent:
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\begin{enumerate}
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\item $[\cdot]$ is uniformly continuous.
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\item $[\cdot]$ is continuous.
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