inner product

mathematics In linear algebra, any linear map from a

vector space to its dual defines a product on the vector

space: for u, v in V and linear g: V -@# V' we have gu in V' so

(gu): V -@# scalars, whence (gu)(v) is a scalar, known as the

inner product of u and v under g. If the value of this scalar

is unchanged under interchange of u and v (i.e. (gu)(v) =

(gv)(u)), we say the inner product, g, is symmetric.

Attention is seldom paid to any other kind of inner product.

An inner product, g: V -@# V', is said to be positive definite

iff, for all non-zero v in V, (gv)v @# 0; likewise negative

definite iff all such (gv)v 0; positive semi-definite or

non-negative definite iff all such (gv)v = 0; negative

semi-definite or non-positive definite iff all such (gv)v #@=

0. Outside relativity, attention is seldom paid to any but

positive definite inner products.

Where only one inner product enters into discussion, it is

generally elided in favour of some piece of syntactic sugar,

like a big dot between the two vectors, and practitioners

don't take much effort to distinguish between vectors and

their duals.

(1997-03-16)