摘要
由于n——赋范空间L上的n-1个元素x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n-1</sub>（线性无关）,可构成一个n-1维子空间Span{(x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n-1</sub>}=V(x<sub>1</sub>,x<sub>2</sub>,…x<sub>n-1</sub>),从而得商空间L/V(x<sub>1</sub>,…,x<sub>n-1</sub>)用Lx<sub>1</sub>,x<sub>2</sub>,…,x<sub>n-1</sub>表示.再设由L×V（x<sub>1</sub>）×V（x<sub>2</sub>）×…×V（x<sub>n</sub>）上的有界n——线性泛函的全体构成的一个线性赋范空间为L<sup>*</sup>(L,V（x<sub>1</sub>）,…,V(x<sub>n-1</sub>).则我们得到L<sup>*</sup>x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n-1</sub>保距线性同构于L<sup>*</sup>(L,V（x<sub>1</sub>）,…,V(x<sub>n-1</sub>).此外我们还得到n-赋范空间L中任何元x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub>,存在Span{x<sub>1</sub>,…,x<sub>n</sub>}上的有界n——线性泛函F,使‖F‖≤1且F（x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub>）=‖x<sub>1</sub>,x<sub>2</sub>,…,x<sub>n</sub>‖.
<ABSTRACT>Since the n-1 elements x_1, x_2, …, x_(n-1) (linear independent) in n-normed pace L can be formed a n-1 dimensional subspace Span {x_1, x_2, …, x_(n-1)} = V(x_1, x_2, …, x_(n-1)), quotient space L/V(x_1, x_2, …, x_(n-1)) can be represented by Lx_1, …x_(n-1). Again, let L*(L, V(x_1), …, V(x_(n-1))) be a normed space which s formed by the whose bounded linear functional on L×V(x_1)×…×V(x_(n-1)), Then we obtained that the distance preserving linear isomorph of L*x_1, …, x_(n-1) is L*(L,V(x_1), …, V(x_(n-1))). In addition, we also obtained that there exist bounded n-linear functional F on subspace Span {x_1, x_2, …, x_(n-1)}, for any elements x_1, x_2, …, x_n in n-normed Space L, such that ‖F‖≤1, and F(x_1, x_2, …, x_n) = ‖x_1, x_2, …, x_n‖
出处
《哈尔滨师范大学自然科学学报》
CAS
1990年第4期20-24,共5页
Natural Science Journal of Harbin Normal University
关键词
n-赋范空间
有界
n-线性泛函
: n——normed space
Bounded n——linear functional
Distance preserving linear isomorph
