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.