Suppose \(\mathbb{K}\) is either \(\mathbb{C}\) or \(\mathbb{R}\). The map \(p\mapsto \lVert x\rVert_p\) is monotonically decreasing on \((0,\infty )\) for every \(x\in \mathbb{K}^n\).

Proof

Let \(q \[ \lVert y\rVert_q^q \ge \lVert y\rVert_p^p=1, \]

since each summand is smaller then one This implies

\[ \biggl\lVert \frac{x}{\lVert x\rVert}_p\biggr\rVert_q\ge 1 \]

and by multiplying \(\lVert x\rVert_p\) on both sides (and applying the homogeneity of the norm) we obtain

\[ \lVert x\rVert_q\ge \lVert x\rVert_p. \]

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