Editing Lp space (section)

====Relations between {{math|''p''}}-norms====

The grid distance or rectilinear distance (sometimes called the "[[Manhattan distance]]") between two points is never shorter than the length of the line segment between them (the Euclidean or "as the crow flies" distance). Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm:
<math display="block">\|x\|_2 \leq \|x\|_1 .</math>

This fact generalizes to <math>p</math>-norms in that the <math>p</math>-norm <math>\|x\|_p</math> of any given vector <math>x</math> does not grow with <math>p</math>:
{{block indent | em = 1.5 | text = <math>\|x\|_{p+a} \leq \|x\|_p</math> for any vector <math>x</math> and real numbers <math>p \geq 1</math> and <math>a \geq 0.</math> (In fact this remains true for <math>0 < p < 1</math> and <math>a \geq 0</math> .)}}

For the opposite direction, the following relation between the <math>1</math>-norm and the <math>2</math>-norm is known:
<math display="block">\|x\|_1 \leq \sqrt{n} \|x\|_2 ~.</math>

This inequality depends on the dimension <math>n</math> of the underlying vector space and follows directly from the [[Cauchy–Schwarz inequality]].

In general, for vectors in <math>\Complex^n</math> where <math>0 < r < p:</math>
<math display="block">\|x\|_p \leq \|x\|_r \leq n^{\frac{1}{r} - \frac{1}{p}} \|x\|_p ~.</math>

This is a consequence of [[Hölder's inequality]].