Suppose \(\mathbb{K}\) is either \(\mathbb{R}\) or \(\mathbb{C}\). The \(l^p\)-norm of a vector \(x\in \mathbb{K}^d\) is given by
\[ \lVert x\rVert_p=\Bigl(\sum_{i=1}^{d}\lvert x_i\rvert^p\Bigr)^{1/p} \]if \(p\in (0,\infty )\) and for \(p=\infty\) we set
\[ \lVert x\rVert_\infty = \sup_{i\in \{1,\ldots ,d\}}\lvert x_i\rvert. \]For \(p=0\) we define
\[ \lVert x\rVert_{0}=\lvert \supp x\rvert, \]where \(\supp\) is the support of \(x\) and \(\lvert x\rvert\) is vector with absolute values of \(x\).
Remarks
Norm Property Link to heading
Not all maps are real norms.
- The maps \(\lVert \cdot\rVert_p\) are norms for \(p\in [1,\infty ]\).
- The maps \(\lVert \cdot\rVert_p\) are not norms for \(p\in [0,1)\).
Relations between Different \(l^p\)-Norms Link to heading
- We have \(\lVert x\rVert_p\to \lVert x\rVert_\infty\) as \(p\to \infty\).
- We have \(\lVert x\rVert_p^p\to \lVert x\rVert_0\) as \(p\to 0^+\).
- The map \(p\mapsto \lVert x\rVert_p\) is monotonically decreasing.