Archive for the ‘Mathematics’ Category

\mathcal{R} as an ordered field with lubp
Rational number system (\frac{m}{n} with m,n\in Q) has gaps (e.g., p can not be find in rational set if p^2=2), in spite of the fact that between any two rationals there is another: if r<s with both r, s\in Q, then r<(r+s)/2<s. The real number system, denoted by R, fills these gaps. This is the principle reason for the fundamental role which it plays in analysis. To formalize this property, we have the least-upper-bound property of real system.

As a preliminary, note that R is an ordered set, where r<s (r,s\in R) is defined as that s-r is positive. For an ordered set, we have the definition of bounded set. The least-upper-bound property is stated as follows:

Property 1 least-upper-bound property
If E\subset R, E is not empty, and E is bounded above, then supE exists in R.

Here supE is the least upper bound of E, which is defined that supE is an upper bound of E, and for any \gamma <\text{sup}E , \gamma is not an upper bound of E. It can be proved that an ordered set with least-upper-bound property also has the greatest-lower-bound property, which can be similarly stated.

Another property that is as important as this one is that R is a field, an algebraic structure which defines the axioms for two operations: addition and multiplication. The importance comes from that the arithmetic laws derived from the definition of field.

To sum up, the set R of real numbers is an ordered field which has the least-upper-bound property. For the details of the proof, you may refer to rudin’s book. With such properties, we can prove the unique existence of \sqrt{n} for any n\in N_+.

As an extension to the real number system, we add two symbols,  +\infty, -\infty, such that  -\infty< x <+\infty for every  x\in R. In this case, any unbounded subset E of  R now has sup E=+\infty in the extended real number system.

Basic topology 1: Euclidean space as a metric space
A metric space is a space where distance (or norm) between elements of this space is defined. The properties of distance is not going to be presented here. The vector space  R^k with definitions of inner product and norm is called Euclidean  k- space.

As preliminaries for getting to know metric space, we may have the definitions of countability first. With the definition of distance, we then immediately derive the definition of bounded set in metric space.

The important concepts in metric space include:

Definition 1 Let X be a metric space,
A neighborhood of  p is a set  N_r(p) consisting of all q such that  d(p,q)<r for some r>0.
A point  p is a limit point of the set  E if every neighborhood of p contains a point q\neq p such that  q\in E.
E is closed if every limit point of  E is a point of  E.
A point  p is called an interior point of  E if there is a neighborhood  N of  p such that  N\subset E.
E is open if every point of  E is an interior point of  E.

Based on the above definition, we have the theorem that 1) every neighborhood is an open set. 2) if p is a limit point of E, then every neighborhood of p contains infinitely many points of E.

Next we combine the above definitions with least-upper-bound in real system. Suppose E\subset X and X is a metric space, let E^{\prime} denote the set of all limit points of E, we define the closure of E: \bar{E}=E\cup E^{\prime}. The theorem we have is:

Theorem 1 supE is a limit point of E
Let E be a nonempty set of real numbers which is bounded above, then supE\in \bar{E}. Hence supE\in E if E is closed.

Now we continues the exploration of concepts in Definition 1 (to be cont..)

Basic topology 2: compact subset of a metric space

[The following notes are written in pdf, you may contact me if you think it useful…]

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http://yinlianqian.blog.163.com/blog/static/648875672008222113952182/

长度是怎样炼成的?

点没有长度和面积,为什么由点组成的线和面会具有长度和面积?

“长度”“面积”这些词汇究竟是在怎样的意义上被使用的?

有的时候我们把点的长度叫做零,有的时候叫做无穷小,这两个称呼是不是都有道理?
无穷个零相加是不是还得零?(其实和第一个问题是一个意思,无穷个点怎么加成线段
的?)

等等等等。

当然,小乐的问题是着眼于哲学,而我的回答将会着眼于数学,——我不是学哲学的,
但是大概也知道在哲学上这些词汇常常导致混乱的争论,比如芝诺悖论之类。幸运的是
,早在一百年前,通过一大批杰出的数学家的努力,以上这些问题已经被精确地给出了
解答,这就是在数学中被称为“测度论”的一套理论体系。这里“精确”的意思是说,
这套理论体系完全基于形式逻辑,而且只采用了非常少的公理(下面会陈述之),从而
,在这套理论中不存在任何模糊或者逻辑上模棱两可之处(除了几个需要加以特别说明
的地方=_=!)。换句话说,我们不仅可以认为数学家能够确定无疑的回答以上这些问题
,而且可以认为人类在今天能够确定无疑的回答以上这些问题(在承认那些公理的前提
下)。

不幸的是,这一断言几乎必然会遭到哲学家的反对。一方面是因为哲学家们倾向于每个
人自己创造一组定义,——从我在未名哲学版见过的一系列关于芝诺悖论的讨论来看,
这样的结果是所有的论述最终都流于自说自话。另一方面大概也因为学术壁垒的缘故,
哲学家们大概从来也没有了解过数学家们已经在此问题上做出过的卓越工作,(确实,
很多细节是过于数学化了一点……)。有鉴于此,我答应小乐以尽可能通俗的方式(在
不损害准确性的前提下)大致介绍一下测度论的内容。我想在这个版面上大概还会有不
少别的朋友对此感兴趣吧。
(more…)

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