Abstract: A vector ｜x〉 expressed by the real number x, the creation operator and the ground state of particle number operator is obtained in terms of the eigenvectors and the completeness relation of particle number operator. It is proved that the vector ｜x〉 is the eigenvector of coordinate operator; the integral over coordinate projective operators is equal to the identity operator; and the eigenvectors of coordinate operator are orthogonal. Hence all eigenvectors { ｜x〉 } form a complete and orthogonal set of basic vectors in an infinite-dimensional vector space.

Key words: eigenvectors of particle number operator, eigenvectors of coordinate operator, integral for projective operator, completeness of coordinate eigenvectors