On the Definition of a Quadratic Form

Svetozar Kurepa

In the first part of this paper we give a simple proof of the following wellknown theorem [3]: If a function $q:X\to C$ satisfies the parallelogram law and the homogeneity property $q(\lambda x) =|\lambda|^2q(x)\;(\lambda\in C,x\in X)$, then there exists a sesquilinear form $L:X\times X\to C$ such that $q(x)=L(x;x)\quad (x\in X)$. If $X$ is a real vector space then a quadratic form on $X$ is to be defined as a function $q:X\to R$ the complexification $(q_c(q_c(x+iy)=q(x)+q(y); x,y\in X)$ of which has the homogeneity property $$ q_c(łambda z)=|łambda|^2q_c(z)\quad (łambda\in C, z\in X_c=X\times X). $$ In the second part of this paper we continue the study of quadratic forms on modules over algebras studied in [6], [7] and [4]. We assume as in [4] that the algebra $A$ has the identity element and that it as the regularity property: For any $t\in A$ there exists a natural number $n$ such that $t+n$ and $t+n+1$ are invertible in $A$.