 
Summary: Hermitian inner products.
Suppose V is vector space over C and
(·, ·)
is a Hermitian inner product on V . This means, by definition, that
(·, ·) : V × V C
and that the following four conditions hold:
(i) (v1 + v2, w) = (v1, w) + (v2, w) whenever v1, v2, w V ;
(ii) (cv, w) = c(v, w) whenever c C and v, w V ;
(iii) (w, v) = (v, w) whenever v, w V ;
(iv) (v, v) is a positive real number for any v V {0}.
These conditions imply that
(v) (v, w1 + w2) = (v, w1) + (v, w2) whenever v, w1, w2 V ;
(vi) (v, cw) = c(v, w) whenever c C and v, w V ;
(vii) (0, v) = 0 = (v, 0) for any v V .
In view of (iv) and (vii) we may set
v = (v, v) for v V
and note that
(viii) v = 0 v = 0.
We call v the norm of v. Note that
(ix) cv = cv whenever c C and v V .
