Vector
Norms and Matrix Norms

q**Norm**: Let *X
*be a
vector space. ||•|| is a norm if

(i) ||*x*||³0 (positivity);

(ii) ||*x*||=0 if and only if* x* = 0 (positive
definiteness);

(iii) ||*a**x*||=|*a*| ||*x*|| for any scalar *a* (homogeneity);

(iv) ||*x+y*||£||*x*||+||*y*|| (triangle
inequality)

for any *x*Î *X *and *y*Î *X*.

Let *x*Î **C**n. Then we define the vector p-norm of *x* as