First, we consider norms of vectors x in a linear vector space V. A norm on V, denoted by , has to satisfy the following four conditions ( x, y ? V, ? ? (or C)):

1. x ? and x ?0

2. x=0 if and only if x=0

3. ? x= ? x

4. x+ y ? x+ y (triangular inequality).

If V= , the norm is often a so-called H lder norm, defined by

with x _i the components of x. For p=1, 2, and ? we have

respectively.

Two norms _? and _? are called equivalent if two constants c ₁, c ₂ exist, such that for all x ? V

From this we have that a series convergent with respect to one norm is also convergent with respect to an equivalent norm. If V= this implies that convergence considerations are norm independent because any two norms on are equivalent. See, e.g., [39].

Next matrix norms are considered. Let V be the linear space consisting of matrices. A norm on V satisfies the conditions similar to (i), , (iv). A vector norm induces a matrix norm in a natural way as follows:

Often such a norm is...