Asymmetric norm explained

In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.

Definition

X is a function p : X \to [ 0 , +inf ty) that has the following properties:

• Subadditivity, or the triangle inequality:

p(x + y) \leq p(x) + p(y) forall x, y \in X.

• Nonnegative homogeneity:

p(rx) = r p(x) forall x \in X and every non-negative real number r \geq 0 .

• Positive definiteness:

p(x) > 0 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 ≠ 0 , at least one of the two numbers p(x) and p( - x) is not zero.

Examples

\R, the function p given byis an asymmetric norm but not a norm.

In a real vector space

X, the p _{B} of a convex subset B\subseteq X that contains the origin is defined by the formula$$p\_B(x)\; =\; \backslash inf\; \backslash left\backslash \backslash ,$$ for x \in X .This functional is an asymmetric seminorm if B is an absorbing set, which means that c up _{r} r B = X, and ensures that p(x) is finite for each x \in X.

Corresponce between asymmetric seminorms and convex subsets of the dual space

If

B ^{*} \subseteq \R ^{n} is a convex set that contains the origin, then an asymmetric seminorm p can be defined on \R ^{n} by the formula$$p(x)\; =\; \backslash max\_\; \backslash langle\backslash varphi,\; x\; \backslash rangle.$$For instance, if B ^{*} \subseteq \R ^{2} is the square with vertices (\pm 1 ,\pm 1 ), then p is the taxicab norm x = \left(x _{0,} x _{1\right)} \mapsto \left|x _{0\right|} + \left|x _{1\right|.} Different convex sets yield different seminorms, and every asymmetric seminorm on \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 ^{*} \subseteq X ^{*} is a compact convex subset of the dual space X ^{*} that contains the origin, then p(x) = max _{\varphi \in B ^{*}} \varphi(x) is an asymmetric seminorm on X.

References

• Cobzaş. S.. Compact operators on spaces with asymmetric norm. Stud. Univ. Babeş-Bolyai Math.. 51. 2006. 4. 69 - 87. math/0608031 . 2006math......8031C . 0252-1938. 2314639.
• S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Frontiers in Mathematics, Basel: Birkhäuser, 2013; .