Continuous Functions on Real and Complex Normed Linear Spaces
Continuous Functions on Real and Complex Normed Linear Spaces
复制标题
实数和复数赋范线性空间上的连续函数
DOI：
--
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
N. Endou
中科院分区：
文献类型：
--
作者：
N. Endou
这里使用的符号和术语在下列文件中介绍：[25]、[28]、[29]、[4]、[30]、[6]、[14]、[5]、[2]、[24]、[10]、[26]、[27]、[19]、[15]、[12]、[13]、[11]、[31]、[20]、[3]、[1]、[16]、[21]、[17]、[23]、[7]、[8]、[22]、[18]和[9]。为了简单起见，我们使用如下约定：n表示自然数，r，S表示实数，z表示复数，C_1，C_2，C_3表示复赋范空间，R_1表示实赋范空间。设C4是复线性空间，S1是C4的序列。函子−s1产生C4序列并由：(def.1)对于每n个保持(−s1)(N)=−s1(N)。下列命题为真：(1)对于所有序列S2，C1的S3成立S2−S3=S2+−S3。(2)对于C1的每个序列S1，存在−S1=(−1C)·S1。让我们考虑C2，C3，设f是从C2到C3的部分函数。产生从C2的载体到R的部分函数的函子‖f‖由下式定义：(def.2)DOM‖f‖=DOM f，且对于C2的每个点c，使得c∈DOM‖f‖保持‖f‖(C)=‖FC‖。让我们考虑c1，r1，设f是从c1到r1的部分函数。从C 1到R的载体产生部分函数的函子‖f‖定义如下：(def.3)DOM‖f‖=DOM f，且对于C1中的每个点c，使得c∈DOM‖f‖保持‖f‖(C)=‖FC‖。
The notation and terminology used here are introduced in the following papers: [25], [28], [29], [4], [30], [6], [14], [5], [2], [24], [10], [26], [27], [19], [15], [12], [13], [11], [31], [20], [3], [1], [16], [21], [17], [23], [7], [8], [22], [18], and [9]. For simplicity, we use the following convention: n denotes a natural number, r, s denote real numbers, z denotes a complex number, C1, C2, C3 denote complex normed spaces, and R1 denotes a real normed space. Let C4 be a complex linear space and let s1 be a sequence of C4. The functor −s1 yields a sequence of C4 and is defined by: (Def. 1) For every n holds (−s1)(n) = −s1(n). The following propositions are true: (1) For all sequences s2, s3 of C1 holds s2 − s3 = s2 + −s3. (2) For every sequence s1 of C1 holds −s1 = (−1C) · s1. Let us consider C2, C3 and let f be a partial function from C2 to C3. The functor ‖f‖ yielding a partial function from the carrier of C2 to R is defined by: (Def. 2) dom‖f‖ = dom f and for every point c of C2 such that c ∈ dom‖f‖ holds ‖f‖(c) = ‖fc‖. Let us consider C1, R1 and let f be a partial function from C1 to R1. The functor ‖f‖ yielding a partial function from the carrier of C1 to R is defined as follows: (Def. 3) dom‖f‖ = dom f and for every point c of C1 such that c ∈ dom‖f‖ holds ‖f‖(c) = ‖fc‖.