  Let F be a field complete for a real valuation. It is a standard result in
valuation theory that a finite extension of F admits a valuation basis if and
only if it is without defect. We show that even otherwise, one can construct
bases in which the discrepancy between measuring valuation of an element versus
on the components in its basis decomposition can be made arbitrarily small. The
key step is to verify this for extensions of degree equal to the characteristic
by a direct calculation.
