asymmetric norm

An asymmetric norm on a real vector space $X$ is a function $p\; :\; X\; \backslash to\; [0,\; +\backslash infty)$ that has the following properties:

• Subadditivity, or the triangle inequality: $p(x\; +\; y)\; \backslash leq\; p(x)\; +\; p(y)\; \backslash text\{\; for\; all\; \}\; x,\; y\; \backslash in\; X.$
• Nonnegative homogeneity: $p(rx)\; =\; r\; p(x)\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; X$ and every non-negative real number $r\; \backslash geq\; 0.$
• Positive definiteness: $p(x)\; >\; 0\; \backslash text\{\; unless\; \}\; x\; =\; 0$

Asymmetric norms differ from norms in that they need not satisfy the equality $p(-x)\; =\; p(x).$

If the condition of positive definiteness is omitted, then $p$ is an asymmetric seminorm. A weaker condition than positive definiteness is non-degeneracy: that for $x\; \backslash neq\; 0,$ at least one of the two numbers $p(x)$ and $p(-x)$ is not zero.

On the real line $\backslash R,$ the function $p$ given by

is an asymmetric norm but not a norm.

In a real vector space $X,$ the {{em|Minkowski functional}} $p\_B$ of a convex subset $B\backslash subseteq\; X$ that contains the origin is defined by the formula

$$p\_B(x)\; =\; \backslash inf\; \backslash left\backslash \{r\; \backslash geq\; 0:\; x\; \backslash in\; r\; B\; \backslash right\backslash \}\backslash ,$$ for $x\; \backslash in\; X$.

This functional is an asymmetric seminorm if $B$ is an absorbing set, which means that $\backslash bigcup\_\{r\; \backslash geq\; 0\}\; r\; B\; =\; X,$ and ensures that $p(x)$ is finite for each $x\; \backslash in\; X.$

If $B^*\; \backslash subseteq\; \backslash R^n$ is a convex set that contains the origin, then an asymmetric seminorm $p$ can be defined on $\backslash R^n$ by the formula

$$p(x)\; =\; \backslash max\_\{\backslash varphi\; \backslash in\; B^*\}\; \backslash langle\backslash varphi,\; x\; \backslash rangle.$$

For instance, if $B^*\; \backslash subseteq\; \backslash R^2$ is the square with vertices $(\backslash pm\; 1,\backslash pm\; 1),$ then $p$ is the taxicab norm $x\; =\; \backslash left(x\_0,\; x\_1\backslash right)\; \backslash mapsto\; \backslash left|x\_0\backslash right|\; +\; \backslash left|x\_1\backslash right|.$ Different convex sets yield different seminorms, and every asymmetric seminorm on $\backslash R^n$ can be obtained from some convex set, called its dual unit ball. Therefore, asymmetric seminorms are in one-to-one correspondence with convex sets that contain the origin. The seminorm $p$ is

• positive definite if and only if $B^*$ contains the origin in its topological interior,
• degenerate if and only if $B^*$ is contained in a linear subspace of dimension less than $n,$ and
• symmetric if and only if $B^*\; =\; -B^*.$

More generally, if $X$ is a finite-dimensional real vector space and $B^*\; \backslash subseteq\; X^*$ is a compact convex subset of the dual space $X^*$ that contains the origin, then $p(x)\; =\; \backslash max\_\{\backslash varphi\; \backslash in\; B^*\}\; \backslash varphi(x)$ is an asymmetric seminorm on $X.$

• {{annotated link|Finsler manifold}}
• {{annotated link|Minkowski functional}}

• {{cite journal|last=Cobzaş|first=S.|title=Compact operators on spaces with asymmetric norm|journal=Stud. Univ. Babeş-Bolyai Math.|volume=51|year=2006|issue=4|pages=69–87|arxiv=math/0608031 |bibcode=2006math......8031C |issn=0252-1938|mr=2314639}}
• S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Frontiers in Mathematics, Basel: Birkhäuser, 2013; {{ISBN|978-3-0348-0477-6}}.

Category:Linear algebra

Category:Norms (mathematics)

{{Functional analysis}}

{{Topological vector spaces}}

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