The Determinant Construction And Minkowski Construction Of N-Normed Linear Space

Based on theses《n-normed Space)) of Yan Gexing and《n-Banach Space and smooth-ness of 2-normed space》of Yu Jianbo, we get some preliminary study about n-normed space as follows:First, we make some preliminary study on the specific construction of n-normed space.A n-norm not only can be induced by the norm of a normed space,but also can be pro-duced by a inner-product.Generally speaking,a n-norm can also be induced by a Hamel basis of a linear space. Second,we can give a set A which satisfies the symmetry,partially absolute convex property,compatible property and absorbing property trending to zero property in the product space Xn.Thus A can generate the Minkowski functional,which can be called the n-seminorm generated by A.Moreever we have proved that the n-semiform generated by A becomes a n-norm if and only if A is separated. Third,we also get three fundamen-tal results,such as the n-norm induced by the norm of the dual space of a normed space X and the dual n-norm of X are the same;the n-norm induced by the inner-product of a inner-product space is the same as the n-norm induced by the norm which is induced by the inner-product;for a more general linear space X,the n-norm induced by a Hamel basis H is the same as the n-norm induced by the norm generated by H.Consequently,we make some investigation upon the aspect of topology in the n-normed space,get a conclusion such as any finite dimensional n-normed space is a n-Banach space,give two characterizations upon when a normed space becomes a n-Banach space,and get a "n-norm and norm" comparison theorem.