Editing Lp space (section)

====When {{math|0 < ''p'' < 1}}====

[[Image:Astroid.svg|thumb|right|[[Astroid]], unit circle in <math>p = \tfrac{2}{3}</math> metric]]
In <math>\Reals^n</math> for <math>n > 1,</math> the formula
<math display="block">\|x\|_p = \left(|x_1|^p + |x_2| ^p + \cdots + |x_n|^p\right)^{1/p}</math>
defines an absolutely [[homogeneous function]] for <math>0 < p < 1;</math> however, the resulting function does not define a norm, because it is not [[subadditivity|subadditive]]. On the other hand, the formula
<math display="block">|x_1|^p + |x_2|^p + \dotsb + |x_n|^p</math>
defines a subadditive function at the cost of losing absolute homogeneity. It does define an [[F-space|F-norm]], though, which is homogeneous of degree <math>p.</math>

Hence, the function
<math display="block">d_p(x, y) = \sum_{i=1}^n |x_i - y_i|^p</math>
defines a [[metric space|metric]]. The [[metric space]] <math>(\Reals^n, d_p)</math> is denoted by <math>\ell_n^p.</math>

Although the <math>p</math>-unit ball <math>B_n^p</math> around the origin in this metric is "concave", the topology defined on <math>\Reals^n</math> by the metric <math>B_p</math> is the usual vector space topology of <math>\Reals^n,</math> hence <math>\ell_n^p</math> is a [[locally convex]] topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of <math>\ell_n^p</math> is to denote by <math>C_p(n)</math> the smallest constant <math>C</math> such that the scalar multiple <math>C \, B_n^p</math> of the <math>p</math>-unit ball contains the convex hull of <math>B_n^p,</math> which is equal to <math>B_n^1.</math> The fact that for fixed <math>p < 1</math> we have
<math display="block">C_p(n) = n^{\tfrac{1}{p} - 1} \to \infty, \quad \text{as } n \to \infty</math>
shows that the infinite-dimensional sequence space <math>\ell^p</math> defined below, is no longer locally convex.{{citation needed|date=November 2015}}