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Quasi-Norm

A quasi-norm is a nonnegative function q:V->[0,infty) on a vector space V over the real numbers or complex numbers that satisfies q(x)=0 iff x=0, absolute homogeneity q(lambdax)=|lambda|q(x), and a relaxed triangle inequality

q(x+y)<=C(q(x)+q(y))

for some fixed constant C>=1 (Hussain et al. 2013, p. 3). It is a norm when the triangle inequality holds with C=1.

For 0<p<1 and a positive integer n, the expression q(x)=(sum_(i=1)^(n)|x_i|^p)^(1/p) is a quasi-norm on R^n or C^n, but not a norm when n>=2. The corresponding integral means of polynomials on the unit circle occur in the Baernstein quasi-norm monotonicity conjecture.

See also

Baernstein Quasi-Norm Monotonicity Conjecture, Norm, Polynomial Norm, Triangle Inequality

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References

Hussain, N.; Salimi, P.; and Al-Mezel, S. "Coupled Fixed Point Results on Quasi-Banach Spaces with Application to a System of Integral Equations." Fixed Point Th. Appl. 2013, Article 261, 2013. https://doi.org/10.1186/1687-1812-2013-261.

Cite this as:

Weisstein, Eric W. "Quasi-Norm." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Quasi-Norm.html

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