Introduction to Banach and Hilbert Spaces - Part I
Introduction to Banach and Hilbert Spaces - Part I
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巴拿赫空间和希尔伯特空间简介 - 第一部分
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发表时间：
1991
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通讯作者：
Jan Popio
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作者：
Jan Popio
文章[5],[18],[22],[3],[4],[1],[10],[8],[9],[7],[15],[2],[23],[16],[17],[14],[21],[20],[19],[13],[11],[12],[6]为本文提供的符号和术语。为简单起见，我们遵循以下规则：X是实酉空间，X是X的一个点，g是X的一个点，a、r是实数，M是实数，s1、s2、s3、s4是X的数列，N1是自然数的递增数列，k、n、M是自然数。让我们考虑X， s1。我们说s1是柯西序列当且仅当：（定义1）对于每一个r使得r > 0存在k使得对所有n， m使得n≥k和m≥k满足ρ(s1(n), s1(m)) < r。我们可以证明以下命题：(1)如果s1是常数，则s1是柯西序列。(2) s1是柯西序列当且仅当对于每一个r使得r > 0存在k使得对所有n， m使得n≥k和m≥k满足‖s1(n)−s1(m)‖< r。(3)如果s2是柯西序列，s3是柯西序列，则s2 + s3是柯西序列。(4)若s2为柯西序列，s3为柯西序列，则s2−s3为柯西序列。(5)如果s1是柯西序列，则a·s1是柯西序列。(6)如果s1是柯西序列，则- s1是柯西序列。
The articles [5], [18], [22], [3], [4], [1], [10], [8], [9], [7], [15], [2], [23], [16], [17], [14], [21], [20], [19], [13], [11], [12], and [6] provide the notation and terminology for this paper. For simplicity we follow the rules: X is a real unitary space, x is a point of X, g is a point of X, a, r are real numbers, M is a real number, s1, s2, s3, s4 are sequences of X, N1 is an increasing sequence of naturals, and k, n, m are natural numbers. Let us consider X, s1. We say that s1 is a Cauchy sequence if and only if: (Def.1) for every r such that r > 0 there exists k such that for all n, m such that n ≥ k and m ≥ k holds ρ(s1(n), s1(m)) < r. One can prove the following propositions: (1) If s1 is constant, then s1 is a Cauchy sequence. (2) s1 is a Cauchy sequence if and only if for every r such that r > 0 there exists k such that for all n, m such that n ≥ k and m ≥ k holds ‖s1(n) − s1(m)‖ < r. (3) If s2 is a Cauchy sequence and s3 is a Cauchy sequence, then s2 + s3 is a Cauchy sequence. (4) If s2 is a Cauchy sequence and s3 is a Cauchy sequence, then s2 − s3 is a Cauchy sequence. (5) If s1 is a Cauchy sequence, then a · s1 is a Cauchy sequence. (6) If s1 is a Cauchy sequence, then −s1 is a Cauchy sequence.