is symmetric and positive definite (with forming an orthonormal basis). Also, for any , is real, hence equal to . Thus we have a norm Since the real numbers commute with all quaternions, we have the multiplicative property . In particular, the unit quaternions (also known as , , or )...
Nevertheless, we have a nice result of Hunter that gives positive definiteness for all even degrees . In fact, a modification of his argument gives a little bit more: Theorem 1 Let , let be even, and let be reals. (i) (Positive definiteness) One has , with strict inequality unless ....
for all . In some applications, one also wants to impose positive definiteness, which asserts that for all . These hypotheses are sufficient in the case when is bounded, and in particular when is finite dimensional. However, as it turns out, for unbounded operators these conditions are not,...
Note that the coefficients appearing in (2) do not depend on the final number of variables . We may therefore abstract the role of from the law (2) by introducing the real algebra of formal sums where for each , is an element of (with only finitely many of the being non-zero), and...