The concept of numerical index of a Banach space X was introduced in 1968 by Lumer in a talk at the North British Functional Analysis Seminar. It is a constant n(X) between 0 and 1 relating the norm of an operators on X with its numerical radius. The extreme cases n(X)=0 (which is only possible for real spaces) and n(X)=1 are of particular importance. Some of the most intriguing results and open problems in this field appear in the setting of finite dimensional spaces.

Very recently the definition was adapted to homogeneous polynomials on X and thus the numerical indices of higher order, n^{(k)}(X) were introduced. A wealth of results that are true for the numerical index fail, many times unexpectedly, for the higher order indices. I will be looking at some of these results, concentrating especially on real finite dimensional spaces.