Definition:Equivalence of Norms

Definition

Let $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ be norms on a vector space $V$.

$\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are equivalent if and only if there exist real constants $c$ and $C$ such that:

$\forall \mathbf x \in V: c \norm {\mathbf x}_1 \le \norm {\mathbf x}_2 \le C \norm {\mathbf x}_1$

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): equivalent norms
• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.4$: Normed and Banach spaces. Sequences in a normed space; Banach spaces