meaning of inner product

1. inner product 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 guv 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. guv = gvu, 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, gvv > 0; likewise negative definite iff all such gvv < 0; positive semi-definite or non-negative definite iff all such gvv >= 0; negative semi-definite or non-positive definite iff all such gvv <= 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 dont take much effort to distinguish between vectors and their duals.
2. a real number a scalar that is the product of two vectors

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