Proposition 14.1.9.label Let $K \in \RC$ and $d \in \natp$, then every norm on $K^{d}$ is equivalent to each other.
Proof. Let $[\cdot]: K^{d} \to [0, \infty)$ be an arbitrary and
\[\norm{\cdot}_{1}: K^{d} \to [0, \infty) \quad x \mapsto \sum_{j = 1}^{n} |x_{j}|\]
then the identity $(K^{d}, \norm{\cdot}_{1}) \to (K^{d}, [\cdot])$ is continuous. Let $S = \bracsn{x \in K^d|\ \norm{x}_1 = 1}$, then $S$ is compact with respect to $\norm{\cdot}_{1}$ and hence compact with respect to $[\cdot]$. By continuity of $[\cdot]$, $\min_{x \in S}[x]$ exists and is non-zero.$\square$
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