stylistic:tone
When 0 < p < 1

In R n {\displaystyle \mathbb {R} ^{n}} for n > 1, {\displaystyle n>1,} the formula ‖ x ‖ p = (| x 1 | p + | x 2 | p + ⋯ + | x n | p) 1 / p {\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots +|x_{n}|^{p}\right)^{1/p}} defines an absolutely homogeneous function for 0 < p < 1; {\displaystyle 0<p<1;} however, the resulting function does not define a norm, because it is not subadditive. On the other hand, the formula | x 1 | p + | x 2 | p + ⋯ + | x n | p {\displaystyle |x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}} defines a subadditive function at the cost of losing absolute homogeneity. It does define an F-norm, though, which is homogeneous of degree p. {\displaystyle p.}

Hence, the function d p (x, y) = ∑ i = 1 n | x i − y i | p {\displaystyle d_{p}(x,y)=\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}} defines a metric. The metric space (R n, d p) {\displaystyle (\mathbb {R} ^{n},d_{p})} is denoted by ℓ n p. {\displaystyle \ell _{n}^{p}.}

Although the p {\displaystyle p} -unit ball B n p {\displaystyle B_{n}^{p}} around the origin in this metric is "concave", the topology defined on R n {\displaystyle \mathbb {R} ^{n}} by the metric B p {\displaystyle B_{p}} is the usual vector space topology of R n, {\displaystyle \mathbb {R} ^{n},} hence ℓ n p {\displaystyle \ell _{n}^{p}} is a locally convex topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of ℓ n p {\displaystyle \ell _{n}^{p}} is to denote by C p (n) {\displaystyle C_{p}(n)} the smallest constant C {\displaystyle C} such that the scalar multiple C B n p {\displaystyle C\,B_{n}^{p}} of the p {\displaystyle p} -unit ball contains the convex hull of B n p, {\displaystyle B_{n}^{p},} which is equal to B n 1. {\displaystyle B_{n}^{1}.} The fact that for fixed p < 1 {\displaystyle p<1} we have C p (n) = n 1 p − 1 → ∞, as n → ∞ {\displaystyle C_{p}(n)=n^{{\tfrac {1}{p}}-1}\to \infty,\quad {\text{as }}n\to \infty } shows that the infinite-dimensional sequence space ℓ p {\displaystyle \ell ^{p}} defined below, is no longer locally convex.