R^n上一个连续函数空间的性质

THE PROPERTIES OF A SPACE OF CONTINUOUS FUNCTIONS ON R^n

设W_k={x=(x_1,x_2…,x_n).k≤ρ(x,0)≤(k±1)},C,(R^n)={f;f∈C(R^n),‖f‖=(?)|f(x)|<+∞},φ(R^1)={f:f∈C_1(R^1),f在[k,k+1]上线性,k取整数}。本文讨论了C_1(R^n)中之收敛性。证明了C_1(R^n)为可分的Banach空间,[φ(R^1)]线性同构同胚于l~∞。

Let W_k={(x_1,x_2,…x_n);k^2≤x_1~1+x_2~2+…+x_n^2≤(k+1)~2},C_1(R^n)={f:f∈C(R^n),■|f(x)|<+∞},φ(R^1)={f:f∈C_1(R^1),f is linear on[k,k+1],k is integer}.In this paper,we discuss the convergence of theclements in C_1(R^n)and prove that C_1(R^n)is a separable Banach Space,l and[φ(R^1)]~*,the dual space of φ(R^1),are linear isomorphic and homeomorphic.