Английская Википедия:Asymmetric norm
Шаблон:Short description In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.
Definition
An asymmetric norm on a real vector space <math>X</math> is a function <math>p : X \to [0, +\infty)</math> that has the following properties:
- Subadditivity, or the triangle inequality: <math>p(x + y) \leq p(x) + p(y) \text{ for all } x, y \in X.</math>
- Nonnegative homogeneity: <math>p(rx) = r p(x) \text{ for all } x \in X</math> and every non-negative real number <math>r \geq 0.</math>
- Positive definiteness: <math>p(x) > 0 \text{ unless } x = 0</math>
Asymmetric norms differ from norms in that they need not satisfy the equality <math>p(-x) = p(x).</math>
If the condition of positive definiteness is omitted, then <math>p</math> is an asymmetric seminorm. A weaker condition than positive definiteness is non-degeneracy: that for <math>x \neq 0,</math> at least one of the two numbers <math>p(x)</math> and <math>p(-x)</math> is not zero.
Examples
On the real line <math>\R,</math> the function <math>p</math> given by <math display="block">p(x) = \begin{cases}|x|, & x \leq 0; \\ 2 |x|, & x \geq 0; \end{cases}</math> is an asymmetric norm but not a norm.
In a real vector space <math>X,</math> the Шаблон:Em <math>p_B</math> of a convex subset <math>B\subseteq X</math> that contains the origin is defined by the formula <math display="block">p_B(x) = \inf \left\{r \geq 0: x \in r B \right\}\,</math> for <math>x \in X</math>. This functional is an asymmetric seminorm if <math>B</math> is an absorbing set, which means that <math>\bigcup_{r \geq 0} r B = X,</math> and ensures that <math>p(x)</math> is finite for each <math>x \in X.</math>
Corresponce between asymmetric seminorms and convex subsets of the dual space
If <math>B^* \subseteq \R^n</math> is a convex set that contains the origin, then an asymmetric seminorm <math>p</math> can be defined on <math>\R^n</math> by the formula <math display="block">p(x) = \max_{\varphi \in B^*} \langle\varphi, x \rangle.</math> For instance, if <math>B^* \subseteq \R^2</math> is the square with vertices <math>(\pm 1,\pm 1),</math> then <math>p</math> is the taxicab norm <math>x = \left(x_0, x_1\right) \mapsto \left|x_0\right| + \left|x_1\right|.</math> Different convex sets yield different seminorms, and every asymmetric seminorm on <math>\R^n</math> 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 <math>p</math> is
- positive definite if and only if <math>B^*</math> contains the origin in its topological interior,
- degenerate if and only if <math>B^*</math> is contained in a linear subspace of dimension less than <math>n,</math> and
- symmetric if and only if <math>B^* = -B^*.</math>
More generally, if <math>X</math> is a finite-dimensional real vector space and <math>B^* \subseteq X^*</math> is a compact convex subset of the dual space <math>X^*</math> that contains the origin, then <math>p(x) = \max_{\varphi \in B^*} \varphi(x)</math> is an asymmetric seminorm on <math>X.</math>
See also
References
- Шаблон:Cite journal
- S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Frontiers in Mathematics, Basel: Birkhäuser, 2013; Шаблон:ISBN.
Шаблон:Functional analysis Шаблон:Topological vector spaces
