The definition of a quadratic form.(English)Zbl 0144.02003

Let $$V$$ be a vector space, of finite or infinite dimension, over a field $$F$$, with characteristic $$\ne 2$$; then the author seeks a new definition of a quadratic form on $$V$$. He wants the definition to be intrinsic, and so excludes the classical definition, which reduces to the elementary one in terms of coefficients and coordinates when $$V$$ has finite dimension. He thinks it inappropriate to define a quadratic form, which is a function of one variable, in terms of a bilinear form involving two variables. He finds it unexpectedly difficult to avoid doing so. For example, $$Q = Q(x)$$, $$x$$ in $$V$$, may satisfy the two identities $$Q(v + w) + Q(v - w) = 2Q(v) + 2Q(w)$$ and $$Q(\lambda x) = \lambda^2Q(x)$$ $$(\lambda$$ in $$F)$$, and yet not be a quadratic form.
The reviewer concludes, from this and other interesting results, that it is best to define quadratic in terms of bilinear forms.
Reviewer: G. L. Watson

MSC:

 15A63 Quadratic and bilinear forms, inner products