Theorem 2.5 . [00H9]

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Theorem 2.5.

Let $V$ be a vector space over $k$ and $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ be complete norms on $V$ . If there exists $C>0$ such that $\lVert\mathord{\cdot}\rVert_{2}\leq C\lVert\mathord{\cdot}\rVert_{1}$ , then the norms $\lVert\mathord{\cdot}\rVert_{1}$ and $\lVert\mathord{\cdot}\rVert_{2}$ are equivalent.