Metric spaces

度量空间

及相关的一些知识点

Definition 6.1.5 (Convergence of sequences).

Let ε > 0 be a real number, and let L be a real number. A sequence $(a_n)^\infty_{n={\color{red} N}}$ of real numbers is said to be ε-close to L iff a_n is ε-close to L for every n ≥ N, i.e., we have |a_n − L| ≤ ε for every n ≥ N. We say that a sequence $(a_n)^\infty_{n={\color{Magenta} m}}$ is eventually ε-close to L iff there exists an N ≥ m such that $(a_n)^\infty_{n={\color{Red} N}}$ is ε-close to L. We say that a sequence $(a_n)^\infty_{n={\color{Magenta} m}}$ converges to L iff it is eventually ε-close to L for every real ε > 0.

定义 6.1.5 序列的收敛

设ε > 0 是个实数，并且L是实数。一个实数序列 $(a_n)^\infty_{n={\color{red} N}}$ 叫作是ε-接近 L 的当且仅当对于每个n ≥ N, a_n都是 ε-接近L的，也就是说，对于每个n ≥ N，|a_n − L| ≤ ε。我们说序列 $(a_n)^\infty_{n={\color{Magenta} m}}$ 的终极ε-接近于 L的，当且仅当 存在 N ≥ m，使得 $(a_n)^\infty_{n={\color{Red} N}}$ 是 ε-接近于 L的。我们说序列 $(a_n)^\infty_{n={\color{Magenta} m}}$ 收敛到 L，当且仅当对于每个实数 ε > 0，它是终极 ε- 接近于L的。

Examples 6.1.6. The sequence

0.9, 0.99, 0.999, 0.9999, . . .

is 0.1-close to 1, but is not 0.01-close to 1, because of the first element of the sequence. However, it is eventually 0.01-close to 1. In fact, for every real ε > 0, this sequence is eventually ε-close to 1, hence is convergent to 1.

是0.1-接近于1的，但不是0.01接近于1的，这取决于序列的第一项。但是它是终极0.01-接近于1的。事实上，对于每个 ε > 0，这个序列都是终极ε- 接近于1的，所以，它收敛到1.

我个人的理解终极收敛，N ≥ m，可假设 N = 2 , m = 1，

0.99, 0.999, 0.999 9, . . . 是0.01-接近于于1，因为从第一项开始，所有的项与1的差的绝对值都小于等于0.01.

0.9, 0.99, 0.999, 0.999 9, . . . 是0.1-接近于1，因为从第一项开始，所有的项与1的差的绝对值都小于等于0.1。

但不是0.01-接近于1的，因为它第一项0.9与1的差的绝对值0.1大于0.01，但是它是终极0.01-接近于1 的，因为它从第二项开始0.99与1的差的绝对值小于等于0.01，也就是说“最终”与1的差的绝对值小于等于0.01。可以这么理解，序列的第一项与1的差距稍微大点，但是没关系，最终它是接近于1的，因为从第二项开始，就开始满足于与1的差的绝对值小于等于0.01。所以照着这个思路，这个序列最终会0.01-，0.001-，0.0001-，等等接近于1的，所以它收敛到1。

总之，终极接近，强调“最终”或“趋势”。

In Definition 6.1.5 we defined what it meant for a sequence $(a_n)^\infty_{n={\color{Magenta} m}}$ of real numbers to converge to another real number x; indeed, this meant that for every ε > 0, there exists an N ≥ m such that |x − x_n| ≤ ε for all n ≥ N. When this is the case, we write lim_n→∞ x_n = x.

我们在6.1.5序列收敛中定义了实数序列 $(a_n)^\infty_{n={\color{Magenta} m}}$ 收敛到一个实数x是什么意思，实际上，这指的是对于每个ε > 0，都存在 N ≥ m，使得对于一切n ≥ N，都有|x − x_n| ≤ ε 。当此事成立时，我们记之为lim_n→∞ x_n = x。

可以这么理解：对于某个实数序列 $(a_n)^\infty_{n={\color{Magenta} m}}$ ，如果对于任意一个很小的正数，只要这个序列从第m项或第m项后的某N项开始与x的差的绝对值都小于这个很小的正数，也就是说序列中的元素有“最终”或“趋势”与x的差的绝对值的小于任意很小的正数。那么， $(a_n)^\infty_{n={\color{Magenta} m}}$ 收敛到实数x，或者lim_n→∞ x_n = x。

Intuitively, when a sequence $(a_n)^\infty_{n={\color{Magenta} m}}$ converges to a limit x, this means that somehow the elements x_n of that sequence will eventually be as close to x as one pleases. One way to phrase this more precisely is to introduce the distance function d(x, y) between two real numbers by d(x, y) := |x − y|. (Thus for instance d(3, 5) = 2, d(5, 3) = 2, and d(3, 3) = 0.) Then we have

Lemma 1.1.1.

Let $(a_n)^\infty_{n={\color{Magenta} m}}$ be a sequence of real numbers, and let x be another real number. Then $(a_n)^\infty_{n={\color{Magenta} m}}$ converges to x if and only if lim_n→∞ d(x_n, x)=0.

直观地说，当序列 $(a_n)^\infty_{n={\color{Magenta} m}}$ 收敛到极限x时，指的是序列元素 x_n 将终极地以任意要求的程度接近于x。一个更精确地描述这个事情的途径是，引入两个实数间的距离函数d(x,y)，它定义为d(x,y) :=|x-y|。

引理 12.1.1

设 $(a_n)^\infty_{n={\color{Magenta} m}}$ 是实数序列，并设x是实数，那么 $(a_n)^\infty_{n={\color{Magenta} m}}$ 收敛到x，当且仅当 lim_n→∞ d(x_n, x)=0

Roughly speaking, a metric space is any space X which has a concept of distance d(x,y) – and this distance should behave in a reasonable manner. More precisely, we have

Definition 1.1.2 (Metric spaces).

A metric space (X, d) is a space X of objects (called points), together with a distance function or metric d: X × X → [0, +∞), which associates to each pair x, y of points in X a non-negative real number d(x,y) ≥ 0.

In many cases it will be clear what the metric d is, and we shall abbreviate (X, d) as just X.

一个粗略的说法，一个度量空间是一个具有距离d(x,y)的概念的任意的集合X，其距离应该有合理的特征。为了更精确说明什么是度量空间，这里给度量空间(X, d)下了个定义。

度量空间(X, d)

一个度量空间(X, d)是一个对象（叫作“点”）的X空间，以及一个距离函数或者度量d: X × X → [0, +∞)，它把X中的点的每对点x, y，指派到一个非负实数d(x, y) ≥0 。简写(X, d)为X.

Supremum

Definition 5.5.10 (Supremum).

Let E be a subset of the real numbers. If E is non-empty and has some upper bound, we define sup(E) to be the least upper bound of E (this is well-defined by Theorem 5.5.9). We introduce two additional symbols, +∞ and −∞. If E is non-empty and has no upper bound, we set sup(E) := +∞; if E is empty, we set sup(E) := −∞. We refer to sup(E) as the supremum of E, and also denote it by sup E.

定义上确界，或者最小上界。

设E是实数系的一个子集。如果E不空且有上界，我们就定义sup(E)为E的最小上界，（根据定理5.59，此定义是成功的）。我们引入两个附加的符号，+∞和 −∞。如果E不是空集，并且没有上界，我们就令sup(E) := +∞；如果E是空集，我们就令sup(E) := -∞。我们把sup(E)叫作E的supremum⁽¹⁾，也记作sup E.

(1) supremum通常译作上确界也译作最小上界（见《数学名词》），这只在sup E ∈ R的情况下运用。

Theorem 5.5.9 (Existence of least upper bound).

Let E be a nonempty subset of R. If E has an upper bound, (i.e., E has some upper bound M), then it must have exactly one least upper bound.

定理 5.5.9 (最小上界的存在性)

设E是R的一个非空子集合，如果E有上界(即E有某个上界M)，那么它恰好有一个最小上界。

Remark 5.5.15.

We can of course talk about lower bounds, and greatest lower bounds, of sets E; the greatest lower bound of a set E is also known as the infimum* of E and is denoted inf(E) or inf E. Everything we say about suprema has a counterpart for infima; we will usually leave such statements to the reader. A precise relationship between the two notions is given by Exercise 5.5.1.

注 5.5.15

我们的当然可以谈论集合E的下界及其最大下界。一个集合E的最大下界也叫作E的infimum*，即下确界，记作inf(E)或者inf E。我们对于上确界所说的每个命题，对于下确界都对应一个相反的命题，把这样的命题留给读者。两个概念之间的精确的联系在习题5.5.1中给出，也见于6.2.

Exercise 5.5.1. Let E be a subset of the real numbers R, and suppose that E has a least upper bound M which is a real number, i.e., M = sup(E). Let −E be the set

−E := {−x : x ∈ E}.

Show that −M is the greatest lower bound of −E, i.e., −M = inf(−E)

习题 5.5.1 设E是实数集R的一个子集。假设E有最小上界M，他是一个实数，即M = sup(E)。令 -E 为集合

−E := {−x : x ∈ E}.

证明 -M 是 -E 的最大下界，即 −M = inf(−E)

*Supremum means “highest” and infimum means “lowest”, and the plurals are suprema and infima. Supremum is to superior, and infimum to inferior, as maximum is to major, and minimum to minor. The root words are “super”, which means “above”, and “infer”, which means “below” (this usage only survives in a few rare English words such as “infernal”, with the Latin prefix “sub” having mostly replaced “infer” in English).

*Supremum 的意思是“最高者”，而infimum的意思是“最低者”。口语形式是suprema和infima。Supremum指的是上方的，infimum指的是下方的，这就像maximum指大的，minmum指小的。词根分别是“super”和“infer”，“super”意指“上方(above)”而”infer”意指“下方(below)”（这个用法只在极少的英语单词中出现，例如“infernal”。在英语中，拉丁语前缀“sub”在绝大多数情况下取代了“infer”）。

Remember that we defined the notion of supremum or least upper bound of a set E of reals; this gave an extended real number sup(E), which was either finite or infinite. We now extend this notion slightly.

记得我们定义了实数集E的supremum或者最小上界的概念；这给出了一个广义实数sup(E)，或者是有限的或者是无限的。我们现在对此概念稍作推广。

Definition 6.2.6 (Supremum of sets of extended reals).

Let E be a subset of R*. Then we define the supremum sup(E) or least upper bound of E by the following rule.⁽¹⁾

(a) If E is contained in R (i.e., +∞ and −∞ are not elements of E), then we let sup(E) be as defined in Definition 5.5.10.
(b) If E contains +∞, then we set sup(E) := +∞.
(c) If E does not contain +∞ but does contain −∞, then we set sup(E) := sup(E\{−∞}) (which is a subset of R and thus falls under case (a)).

We also define the infimum inf(E) of E (also known as the greatest lower bound of E by the formula

inf(E) := − sup(−E)

where −E is the set −E := {−x : x ∈ E}.

定义 6.2.6 广义实数集的上确界

设E是R*的子集合，我们用下面的法则来定义E的上确界或最小上界sup(E)⁽¹⁾

(a) 如果E包含于R（即+∞ 和 −∞ 不是 E 的元素），那么我们令sup(E)由定义5.5.10确定。

(b) 如果E包含有+∞，那么我们令sup(E) := +∞。

我们也定义E的下确界（即最大下界）为

inf(E) := − sup(−E)

(1)定义6.2.6和定义5.5.10的一个重要区别是，R*本身就是有界集。R*的任何子集都有最小上界，即上确界。而按定义5.5.10，不能说无上界的集合有上确界。所以在R中，sup(E) = +∞ 不叫作上确界。

Example 6.2.7.

Let E be the negative integers, together with −∞:

E = {−1, −2, −3, −4,…} ∪ {−∞}.

Then

sup(E) = sup(E\{−∞}) = −1,

while

inf(E) := − sup(−E) = −(+∞) = −∞

例6.2.7

设 E 是负整数及 −∞ 的全体：

E = {−1, −2, −3, −4,…} ∪ {−∞}.

那么

sup(E) = sup(E\{−∞}) = −1

而

inf(E) := − sup(−E) = −(+∞) = −∞

Example 6.2.8.

The set {0.9, 0.99, 0.999, 0.9999,…} has infimum 0.9 and supremum 1. Note that in this case the supremum does not actually belong to the set, but it is in some sense “touching it” from the right.

例6.2.8

集合{0.9, 0.99, 0.999, 0.999 9, . . . }的上确界是1，下确界是0.9。注意这里上确界不属于此集合，但它在一定意义上从右方“触及”此集合。

Example 6.2.9.

The set {1, 2, 3, 4, 5 . . . } has infimum 1 and supremum +∞.

集合{1, 2, 3, 4, 5 . . . } 有下确界 1 和上确界+∞.

Example 6.2.10.

Let E be the empty set. Then

sup(E) = −∞

and

inf(E)=+∞ (why?).

This is the only case in which the supremum can be less than the infimum (why?).

设E是空集，那么

sup(E) = −∞

并且

inf(E)=+∞ （为什么？）

这是supremum小于infimum的唯一情况（为什么？）

One can intuitively think of the supremum of E as follows.

Imagine the real line with +∞ somehow on the far right, and −∞ on the far left. Imagine a piston at +∞ moving leftward until it is stopped by the presence of a set E; the location where it stops is the supremum of E. Similarly if one imagines a piston at −∞ moving rightward until it is stopped by the presence of E, the location where it stops is the infimum of E. In the case when E is the empty set, the pistons pass through each other, the supremum landing at −∞ and the infimum landing at +∞.

可以像下面那样直观想象E的上确界。

想象实直线以某种方式在最右端是 +∞长而在最左端是 −∞ 长。想象一个活塞从 +∞ 处向左移动直到遇到E为止；它停下来的位置就是E的上确界。类似地，如果想象一个活塞从-∞ 处向右移动直到遇到E而停，它停下来的位置就是E的下确界。当E是空集时，两个活塞彼此穿过，上确界落在−∞，而下确界落在+∞。

The following theorem justifies the terminology “least upper bound” and “greatest lower bound”

Theorem 6.2.11.

Let E be a subset of R*. Then the following statements are true.
(a) For every x ∈ E we have x ≤ sup(E) and x ≥ inf(E).
(b) Suppose that M ∈ R* is an upper bound for E, i.e., x ≤ M for all x ∈ E. Then we have

sup(E) ≤ M.

inf(E) ≥ M.

下面的定理为术语“最小上界”和“最大下界”提供了合理的解释

定理 6.2.11

设E是R*的子集合，那么下述各命题成立：

(a) 对于每个 x ∈ E ，我们有 x ≤ sup(E) and x ≥ inf(E)。

(b) 设M是E的上界，即对于一切x ∈ E，x ≤ M，那么我们有sup(E) ≤ M。

Norms and Metrics

Download Norms and Metrics

http://www.u.arizona.edu/~mwalker/econ519/Econ519LectureNotes/Norms&Metrics.pdf

Euclidean Norm

欧几里得范数

$\left \| x \right \| =\sqrt{x_1^2 +x_2^2+...+x_n^2}$

Euclidean metric ( l² metric )

$\begin{align*} {\color{Red} d_{l^2}}((x_1,...,x_n),(y_1,...,y_n))&:=\sqrt{(x_1-y_1)^2 + ... +(x_n-y_n)^2}\\ &=\left ( \sum_{i=1}^n (x_i-y_i)^2 \right )^{1/2} \end{align*}$

if n=2, then d_l²((1,6),(4,2)) = √(3² + 4²) = 5. This metric corresponds to the geometric distance between the two points (x₁, x₂, . . . , x_n ), (y₁, y₂, . . . , y_n ) as given by Pythagoras’ theorem.毕达哥拉斯定理

Euclidean distance between two points x, x’ ∈ Rⁿ

$d(x,x'):=\left \| x,x' \right \|$

Taxicab metric ( l¹ metric )

$\begin{align} {\color{Blue} d_{l^1}}({\color{Red} (x_1,x_2,...,x_n)},{\color{Magenta} (y_1,y_2,...,y_n)})&:=\left | x_1-y_1 \right | +...+ \left | x_n - y_n \right | \\ &=\sum_{i=1}^n \left | x_i-y_i \right | \end{align}$

Because it models the distance a taxi-cab would have to traverse to get from one point to another if the cab was only allowed to move in cardinal directions (north, south, east, west) and not diagonally. 出租车只能沿着东西走向或者南北走向的街道开，不能从房子上或者从房子里开过去（虽然两点间直线最短）。

${\color{Red} d_{l^2}(x,y)} \leq {\color{Blue} d_{l^1}(x,y)} \leq \sqrt {n} d_{l^2} (x,y)$

最简单形式的理解可以是直角三角形两直角边之和大于斜边。

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Metric spaces

Metric spaces

度量空间

及相关的一些知识点

Supremum

Example 6.2.7.

例6.2.7

Example 6.2.8.

例6.2.8

Example 6.2.9.

Example 6.2.10.

Norms and Metrics

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Euclidean Norm

欧几里得范数

Euclidean metric ( l² metric )

Euclidean distance between two points x, x’ ∈ Rⁿ

Taxicab metric ( l¹ metric )

$\begin{align} {\color{Blue} d_{l^1}}({\color{Red} (x_1,x_2,...,x_n)},{\color{Magenta} (y_1,y_2,...,y_n)})&:=\left | x_1-y_1 \right | +...+ \left | x_n - y_n \right | \\ &=\sum_{i=1}^n \left | x_i-y_i \right | \end{align}$

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度量空间

及相关的一些知识点

Supremum

Example 6.2.7.

例6.2.7

Example 6.2.8.

例6.2.8

Example 6.2.9.

Example 6.2.10.

Norms and Metrics

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Euclidean Norm

欧几里得范数

Euclidean metric ( l2 metric )

Euclidean distance between two points x, x’ ∈ Rn

Taxicab metric ( l1 metric )

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Euclidean metric ( l² metric )

Euclidean distance between two points x, x’ ∈ Rⁿ

Taxicab metric ( l¹ metric )