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Given a vector space V, a sesquilinear form on V is a map f : V ×V → C with the properties that if a and b are in C and u, v, w are in V, then:

f(au+bv,w) = af(u,w)+bf(v,w) f(w,au+bv) = a*f(w,u)+b*f(w,v).