On condition numbers of a basis

The condition number κp(x̲) of an ordered basis x̲ of a finite dimensional subspace of an Lp-space, 1⩽p⩽∞, measures the extent to which small relative changes in the basis coefficients lead to small relative changes in the function values, and vice versa. We consider an ordered basis x̲ of an arbitrary finite dimensional normed space and introduce numbers ϰp,r(x̲), 1⩽p,r⩽∞, which control 'near linear dependence' of the basis elements as well as overflow/underflow during computations. These numbers are defined in terms of the norms of the basis elements and the norms of the elements of the ordered dual basis. Optimal scaling strategies are determined. Effects of matrix transformation on these numbers are explored. If x̲ is an ordered basis of an inner product space, then ϰp,r(x̲) can be calculated explicitly in terms of the diagonal entries of the Gram matrix corresponding to x̲ and the diagonal entries of its inverse. In the last section, we define a condition number of x̲ relative to the problem of computing the ordered dual basis, and relate it to ϰp,q(x̲), where (1/p)+(1/q)=1.