CAL1_L2

Equivalence of p-norms

Theorem 1(Theorem).

Any two p-norms are equivalent.

A few simple examples first:

• $\Vert \cdot {\Vert }_1$ and $\Vert \cdot {\Vert }_{\infty }$ are equivalent: $\Vert \cdot {\Vert }_{\infty }\le \Vert \cdot {\Vert }_1\le n\Vert \cdot {\Vert }_{\infty }$.
• $\Vert \cdot {\Vert }_p$ and $\Vert \cdot {\Vert }_{\infty }$ are equivalent: $\Vert \cdot {\Vert }_{\infty }\le \Vert \cdot {\Vert }_p\le n^{1/p}\Vert \cdot {\Vert }_{\infty }$.
• Note that equivalence is an equivalence relation, i.e, it is transitive.
  $\Vert \cdot {\Vert }_1$ and $\Vert \cdot {\Vert }_p$ are equivalent: $\frac{1}{n}\Vert \cdot {\Vert }_1\le \Vert \cdot {\Vert }_p\le n^{1/p}\Vert \cdot {\Vert }_1$.

Thus, all p-norms are equivalent.