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


15A63 Quadratic and bilinear forms, inner products
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