 
Summary: Inner products.
Let V be a vector space.
Definition. We say a function
: V × V R
is an inner product on V if
(i) for each v V the function
V w (v, w) R
is linear;
(ii) for each w V the function
V v (v, w) R
is linear;
(iii) (v, w) = (w, v) for each v, w V and
(iv) (v, v) 0 with equality only if v = 0.
Here's some fancy mathematics terminology that goes with this. Properties (i) and (ii) say that i is
bilinear, property (iii) says that i is symmetric and property (iv) says that i is positive definite.
One often writes
v · w
for (v, w). Keeping in mind (iv), for v V we set
v =
