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Definition: An inner product is a generalization of the dot product. In a vector space, it is a way to multiply vectors together, with the result of this multiplication being a scalar. More precisely, for a real vector space, an inner product <*,*> satisfies the following four properties. Let u, v, and w be vectors and alpha be a scalar, then: 1. <u+v,w>=<u,w>+<v,w>. 2. <alphav,w>=alpha<v,w>. 3. <v,w>=<w,v>. 4. <v,v> >= 0 and equal if and only if v=0.