My linear algebra book seems to give a different definition than Mathworld.com so I'll state it. A scalar product over a vectorial space V is a vectorial real function that to every pair of vectors u, v, associates a real number noted (u|v) satisfying the 4 axioms... 1. 2. 3. 4. A vectorial space of finite dimension with a scalar product is called a euclidean space. My question is the following: I don't like how that definition sounds. Is it equivalent to: "Let V be a vectorial space of finite dimension. If there exists a scalar product function over V, then V is called a euclidean space." ? P.S. does anyone knows a good website that teaches about diagonalisation of hermitian matrixes?