巴拿赫空间和希尔伯特空间

F. Narcowich
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摘要

如果对于每个> 0,存在一个自然数N使得‖vj−vk‖<对于所有j, k≥N,则称序列{vj}是柯西的。每一个收敛序列都是柯西的,但是在赋范线性空间V中存在不收敛柯西序列的例子很多。一个这样的例子是有理数q的集合。数列(1.4,1.41,1.414,…)收敛于√2,它不是一个有理数。如果一个赋范线性空间中的每一个柯西序列都是收敛的,我们就说这个空间是完备的。实数是完全赋范线性空间的一个例子。如果赋范线性空间是完备的,我们就说它是巴拿赫空间。我们称完全内积空间为希尔伯特空间。考虑下面的例子:每一个有限维赋范线性空间都是一个巴拿赫空间。同样,每一个有限维的内积空间都是希尔伯特空间。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Banach Spaces and Hilbert Spaces
A sequence {vj} is said to be Cauchy if for each > 0, there exists a natural number N such that ‖vj−vk‖ < for all j, k ≥ N . Every convergent sequence is Cauchy, but there are many examples of normed linear spaces V for which there exist non-convergent Cauchy sequences. One such example is the set of rational numbers Q. The sequence (1.4, 1.41, 1.414, . . . ) converges to √ 2 which is not a rational number. We say a normed linear space is complete if every Cauchy sequence is convergent in the space. The real numbers are an example of a complete normed linear space. We say that a normed linear space is a Banach space if it is complete. We call a complete inner product space a Hilbert space. Consider the following examples: 1. Every finite dimensional normed linear space is a Banach space. Likewise, every finite dimensional inner product space is a Hilbert space.
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