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所在平台: Udemy |
课程主页: https://www.udemy.com/course/analysis-of-metric-spaces/
课程评论：没有评论
《度量空间分析》课程总结 本课程旨在介绍如何将微积分强大的工具，特别是极限的概念，推广到更一般的度量空间中。欧几里得空间是微积分传统的研究对象，但许多高级分析所使用的空间实际上是度量空间。因此，理解度量空间中的极限概念是掌握高级分析的关键。 课程的核心在于“距离”这一概念的泛化，即如何将“接近”的概念从实数轴推广到更一般的集合。在度量空间中，“点”可以是复杂的对象，例如函数空间中的函数。这种抽象化处理能够帮助数学家们剥离复杂的细节，更清晰地识别可适用的定理和技术，从而把握问题的整体脉络，而非纠结于细枝末节。通过学习度量空间，学习者将获得一种用以理解和解决更广泛数学问题的通用语言。
The methods of calculus are limited to Euclidean spaces. In this course, we show how the incredibly powerful tools of calculus, beginning with the limit concept, can be generalized to so-called metric spaces. Almost every space used in advanced analysis is in fact a metric space, and limits in metric spaces are a universal language for advanced analysis. The basic techniques of calculus were invented for the real line R. What should we do when we want to handle something more general than R? The fundamental notions of calculus begin with the idea that one point of R can be close to another point of R, and this is called "Approximation" or "taking a limit." Metrics are a way to transfer this key notion of being "close to" to a more general setting. The "points" of a metric space can be complicated objects in their own right. For example, they may themselves be functions on some other space. Ideas like this are ubiquitous in advanced mathematics today. One tries to throw away complicated details of the space being considered, and this makes it easier to see which theorem or technique can be applied next. In this way, the mathematician tries to avoid getting overwhelmed by the details, or to say it differently, we try to see the overall forest rather than the trees.
