**定理**. 设是测度空间. 当时, 是一致凸的.

**证明. **首先注意到的情形根据Hanner不等式立刻得到.下面考虑的情形, 根据Hanner 不等式, 我们有

在上式中令, 那么有

注意到对于, 我们有下面的估计:

分为有两种情形：

(1) , 根据(1)式和(2)式可得

因此

(2) ，则

因此

当时, 我们平凡地有

因此可以假设, 那么由(3)式可得

注意到, 因此因为, 因此

因此

因此根据假设

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# 分类：Real analysis

# L^p (1<p<infty) 的一致凸性

# 域扩张

## 代数初步

### 交换环上的代数

# 数论问题的价值

# The Kakeya problem in finite fields

# Ordered field

# Fréchet derivative

### Relation to the Gâteaux derivative

### Higher derivatives

# Invariance of domain and retraction

### Invariacne of domain

### Retract

### Deformation retract and strong deformation retract

# Separation theorems in the plane

### 1 The Jordan separation theorem

### 2 Invariance of domain

### 3 The Jordan curve theorem

# Application of ultrafilters

### Examples 1: generalized limits

### Set systems as quantifiers

# Lattice property of the class of signed measures

**定理**. 设是测度空间. 当时, 是一致凸的.

**证明. **首先注意到的情形根据Hanner不等式立刻得到.下面考虑的情形, 根据Hanner 不等式, 我们有

在上式中令, 那么有

注意到对于, 我们有下面的估计:

分为有两种情形：

(1) , 根据(1)式和(2)式可得

因此

(2) ，则

因此

当时, 我们平凡地有

因此可以假设, 那么由(3)式可得

注意到, 因此因为, 因此

因此

因此根据假设

我们考虑域的一般结构。域论探讨的基本课题是域嵌入，或者说域扩张。我们主要考察代数扩张，采取的角度是系统地利用代数闭包的存在性，对代数扩张尽量广泛的处理。

代数学中出现的许多环结构同时是域上的向量空间：环的加法来自向量空间的加法，而乘法是向量空间上的双线性型。典型的例子是域上的矩阵环。

以下设是非零的交换幺环。所谓的代数都是要幺的结合代数。

定义 环上的代数式一个具有环与-模结构的结合$latex $A，使得环乘法是平衡积，亦即

我以为判断一个数学问题的价值大致有2种途径：一是看它是否有趣，一是看它是否重要。

所谓“有趣”，大多与一个命题出乎意料的程度有关。一般来说，一个表述简单却难以证明或证否的问题通常是有趣的。如果这个问题最终得到证明，事情就更加有趣了：从一片混沌中诞生出简单的图景，提示我们必然有某种值得深入研究的机理存在。此时这个问题开始变得“重要”了：围绕着它，数学家们构筑起理论，试图把他们在解决这个问题的过程中所获得的经验推广到更多的问题上。新的数学产生了。

一个问题的重要性取决于它在我们对现象的理解(即“理论”)中占据何种位置。挡在通衢大道上的石头是谁都想搬开的，躺在路边的石头则不会有多少人注意。当然，如果路边的石头固执地抗拒一切推开它的努力，它将以另一种方式变得“重要”：这种重要性由种种失败中所产生的新数学的多少来衡量。

数论是数学中最特别的分支：我们对整数惊人的无知，几乎所有数论问题都在某种程度上是“有趣”的——正是凭借这一点数论吸引了人类最优秀的头脑。另一方面，除了少数几个经典问题外，似乎很难先验地知道哪些问题在整体图景中是“重要”的：此时我们只能转而求助第2种重要性的定义，希望从中产生尽可能多的数学。

下面是一个粗糙的分类。我们仅给出每一类中最具代表性(往往也最有价值)的例子，并不代表所有同类问题都具有同等价值。横向上看，通常(1)(3)中的问题都是值得珍视的(在数论中并不多见)，(2)(4)的价值次之，充斥整个数论的(5)则不那么重要：

(1)第一类问题包括Riemann猜想和Langlands纲领。它们在2种意义上都是重要的：既处在现代数论的核心，又催生了大量“好的数学”，是整个现代数论前进的定向标。

未解决的Hilbert问题中，第9问题和第12问题(Kronecker’s *Jugendtraum*)都可以归入此类。除了本身的理论价值外，它们还是类域论、复乘理论和Langlands纲领的渊薮。

(2)另一类问题有理论上的重要性，却因为太难或者太偏而没有产生太多主流数学，或者必须借助(1)中的问题才能得到迂回的理解：例子包括Gauss的类数猜想，Artin的原根猜想，等等。如果有人能以“正确的方式”理解它们，则此类问题可能提升为(1)中的问题。

(3)Fermat大定理本身并不重要，但它在第二种意义上极端重要：例如，它催生了Kummer的理想理论，从而建立了代数数论和代数几何的基础。Wiles的证明则增进了对Langlands纲领的理解，由此产生的系列数学工具也极具威力(参见Richard Taylor的工作)。

目前看来，比Fermat大定理更强的abc猜想应该也属于此类。望月新一最近的工作能否像Wiles的工作那样推动整个领域的进步，我们拭目以待。

(4)同样，Goldbach猜想和孪生素数猜想本身也没有太大的重要性(尽管它们是“有趣”的典型例子)。人们因此发展了加性数论(华罗庚的“堆垒数论”)。经典工具(例如筛法)的应用范围狭窄，和Fermat大定理衍生出的数学相比，眼下处在边缘位置。这解释了为什么某些数学家轻视这方面的工作。当然，(4)中的问题也有可能提升到(3)：例如，加性数论最近接受了来自遍历理论的新想法，似有重新回归主流的趋势(参见陶哲轩的工作)，而后者又依赖于从到van der Waerden定理到Szemerédi定理的提升。

Erdős是“趣味主义”的代言人，他提出的猜想大多属于(4)。概率数论(Erdős–Kac定理, etc.)和随机图(Erdős–Rényi模型, etc.)等工作是成功提升到(3)的例子，上面提到的陶哲轩的工作可能使Erdős猜想(若$latex sum 1/a_i$发散，则整数序列$latex {a_i}$中包含任意长的算术级数)获得提升。Ramanujan在模函数方面的工作中，Ramanujan猜想已通过Weil猜想成功提升。古老的同余数问题并无重要性，但它通过与BSD猜想的联系获得了重要性。另一个相对近代的例子是经由Vojta的工作，Roth定理成功融入了算术几何的理论框架。

(5)证否和反例不一定是不重要的(尤其在第二种意义上)：例如，Littlewood证否了Gauss猜想$latex pi(n)<mathrm{Li}(n)$，这增进了我们对$latex zeta$函数的理解，值得划入(4)。在寻找Euler猜想反例的过程中，Elkies和Frye等人发现了椭圆曲线理论的一个意外应用，这有一定的算法价值 (更不要说类似的构造椭圆曲线的方法提供了从谷山-志村猜想推出Fermat大定理的途径)。

Guy的*Unsolved Problems in Number Theory*中收录的问题也不一定是不重要的。事实上，它们中相当大的一部分都有某种程度的重要性。我们已在(2)(4)中提到部分例子，尝鼎一脔，其余可知。

很遗憾，在我看来Guy, F26不属于上述两类，而属于最不重要(也最常见)的一类数论问题：既在整个理论中没有位置，也不太可能产生有意思的数学。反例并不巨大(这意味着问题并不是那么难)，同时，由于找到反例的方式是完全初等的，其潜在的算法价值也相当有限——这还是在不考虑Fuller，Iraids等人已得到好得多的结果的情况下。

【注记】

本文写成之后，豆瓣上的魔术师同学提醒我Guy, F26和素数的Kolmogorov复杂度有关。这样看来，一个Littlewood式的证否原本可能将它提升到(4)。我同意他的看法：“这个解法把他降低到了(5)。……一个昭示如何构造反例，或者证明仅有穷多反例，或者无穷多反例才可算是好的回答。”

The *kakeya problem, *in its best known formulation, is the following. Let be set which contains a translate of every unit segment; equivalently, for every direction , contains a unit line segment parallel to . An -dimensional ball of radius is a simple example of a set with this property, but there are many other such sets, some of which have -dimensional measure .

Can be even smaller than that and have Hausdorff dimension strictly smaller that ?

An *ordered filed *is a field together with a total ordering of its elements that is compatible with the field operations.

An ordered field necessarily has characteristic since the elements necessarily are all distinct$. Thus, an ordered field necessarily contains an infinite number of elements: a finite field cannot be ordered.

Every ordered field contains an ordered subfield that is isomorphic to rational numbers. Any Dedekind complete ordered field is isomorphic to the real numbers. Squares are necessarily non-negative in an ordered filed.

**Definition **A filed together with a total order on is an *ordered field* if the order satisfies the following properties for all and $c$ in :

- if then , and
- if and then .

The *Fréchet derivative *is a derivative defined on Banach spaces, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus variations.

**Definition **Let and be Banach spaces, and be an open subset of . A function is called *Fr échet differentiable *at if there exists a bounded linear operator such that

The limit here is meant in the usual sense of a limit of a function defined on a metric space.

**Definition **A function is called *Gâteaux differentiable *at if has a directional derivative along all direction at . This means that the limit

exists for any choosen vector in , where is is from the scalar filed associated with .

If is Fréchet differentiable at , is is also Gâteaux differentiable there, and the limit is just .

If is a differentiable function at all points in an open subset of , it follows that its derivative

is a function from to the space of all bounded liner operators from to . This function may also have a derivative, the *second order derivative* of , which, by the definition of derivative, will be a map

To make it easier to work with second order derivatives, the space on the right-hand side is identified with the Banach space of all continuous bilinear map from to . An element in is thus identified with in such that for all and in

*Invariance of domain *states:

**Theorem (Invariance of domain) **If is an open subset of and is an injective continuous map, then is open and is a homeomorphism between and .

The conclusion of the theorem can equivalently be formulated as: “ is an open map”. It is of crucial importance that both domain and range of are contained in the Euclidean space of *the same dimension*. The theorem is also not generally true in infinite dimension.

A *retraction* is a continuous mapping from the entire space into a subspace which preserves the position of all points in that space. A *deformation retraction *is a map which captures the idea of *continuously shrinking *a space into a subspace.

**Definition **Let be a topological space and be a subspace of . Then a continuous map

is a *retraction* if the restriction of to is the identity map on ; that is, for all in . Equivalently, denoting by

the inclusion, a retraction is a continuous map such that

A subspace is called a retract of if such a retraction exists.

**Theorem **Let be a Hausdorff space and be a retract of . Then is closed.

*Proof *Let and . Since is Hausdorff, and have disjoint neighborhood and , respectively. Then is a neighborhood of disjoint from . Hence, is closed.

A space is known as an *absolute retract* if for every normal space contains latex X$ is a retract of .

**Definition **A continuous map

is a *deformation retraction *of a space onto a subspace if, for every in and in ,

In other words, a deformation retraction is a homotopy between a retraction and the identity map on . The subspace is called a *deformation retract *of . A deformation retraction is a special case of homotop equivalence.

Our proof of the Jordan curve theorem divides into three parts. The first, which we call the *Jordan separation theorem*, states that a simple closed curve in the plane separates it into at least two components. The second says that an arc in the plane does not separate the plane. And the third, the *Jordan curve theorem* proper, says that a simple closed curve in the plane separates it into precisely two components, of which is the common boundary.

**Lemma 1 **Let be a compact subspace of , let be a point of ; and let be a homeomorphism of with . Suppose is a component of . If does not contain , then is a bounded component of . If contains , then is the bounded component of .

In particular, if has components, then has components.

*Proof * We show first that if is a component of , then is connencted.

Let be the set of components of ; let . Because is locally connected, the set are connected, disjoint open subsets of . Therefore the sets are connected, disjoint, open subsets of , so the sets are the components of .

**Lemma 2 (Nulhomotopy lemma) **Let and be points of . Let be a compact space, and let

be a continuous map. If and lie in the same component of , then is nulhomotopic.

**Definition 1 **If is a connected space and , we say that *separates * if is not conneted; if has components, we say that *separates * *into* *components.*

**Definition 2 **An *arc *is a space homeomorphic to the unit inverval . The *end points of * are the two points and such that and are connected; the other points of are called *interior points *of .

A *simple closed curve *is a space homeomorphic to the unit circle .

**Theorem 1**Suppose , where and are open sets of . Suppose that is path connected, and that . Let and be the inclusion mappings of and , respectively, into . Then the images of the induced homomorphisms

generate .

**Theorem 2 (The Jordan separation theorem) **Let be a simple closed curve in . Then separate .

**Lemma 3 (Homotopy extension lemma) **Let be a space such that is normal. Let be a closed subspace of , and let be a continuous map, where is an open subspace of . If is nulhomotopic, then may be extended to a continuous map that is also nulhomotoptic.

The following lemma is partial converse to the nulhomotopy lemma of the preceding section.

**Lemma 4 (Borsuk lemma) **Let and be points of . Let be a compact space, and let be a continuous injective map. If is nulhomotopic, then and lie in the same component of .

**Theorem 3 (Invariance of domain) **If is an open subset o and is continuous and injective, then is open in and the inverse function is continuous.

The special case of the Seifert-van Kampen theorem that we used in proving the Jordan separation theorem tell us something about the fundamental group of the space in the case where the intersection is path connected. In the next theorem, we examine what happens when is *not *path connected. This result will enable us to complete the proof the Jordan curve theorem.

Now we prove the Jordan curve theorem

**Theorem (The Jordan curve theorem) **Let be a simple closed curve in . Then separates into precisely two components and . Each of the sets and has as its boundary; that is, for .

An *ultrafilter *on a set is a collection of subsets of with the following properties:

- and ;
- is closed under finite intersection;
- if and , then
- fr every , either or .

A trivial example of an ultrafilter is the collection of all sets containing some fixed element of . Such ultrafilters are called *principal*. It is not trivial that there are any non-principal ultrafilters, but this can be proved using Zorn’s lemma.

The following facts about filters, which follow easily from the basic definition, will be used in this post. Let be an ultrafilter on a set .

- If is partitioned into finitely many sets , the precisely one belongs to .
- If and do not belong to then neither does .
- If any finite set belongs to then is a principal filter.

We can think of the process of taking limits of sequence as a linear functional defined on the convergent sequences.

Can we generalize by finding a linear functional that is defined on all *bounded *sequences and not just all convergent on ? In order for it to count as a generalization, we would like to be linear, and we would like to equal whenever is convergent sequence.

If is a non-principal ultrafilter, and is a sequence that takes values in , then we can define a *limit along * as follows. Let be the collection of all subintervals of such that belongs . Then the ultrafilter properties of imply that has all ultrafilter properties but restricted to intervals.

From this it follows that is something like a “principal interval-ultrafilter”. More precisely, it contains all open intervals that contain some particular point . To see this, for each partition into finitely many subintervals of length at most . Then one of these subintervals belongs to . So for every we have an interval of length that belongs to . Now let . Since is closed under intersection, belongs to . Let be the intersection of the closures of the (which are non-empety and nested). If is any open interval containing , then contains some , so belongs to .

Thus ,we have found a number a with the following property: for every , the set belongs to . Moreover, it is easy to see that this is unique. We write it as . It is easy to cheak that is linear.

To see ever more clearly how this ties in with the usual notion of a limit, note that converges to if and only if for every , the set belongs to the cofinite filter.

It is often better to think of a set system as a quantifiers. In particular, if is an ultrafilter then one often finds oneself writing sentences of the form , as we have already seen. But it can be much easier to deal with these sentences if one instead writes . One can read this as “For -almost every “.

Signed measures have values either in or , to avoid the possibility of adding to . If is a signed measure space and is a measurable set, define

The set function and are respectively the *positive, negative *and *total variations *of .

**Theorem **If and are signed measures on a measurable space, there is a signed measure majorizing and and majorized by every other signed measure majorant and .

*Proof *If is a well-defined signed measure, that is if and are not both or both . Let be a maximal positivity set and be a maximal negativity set, for . Define

This sum defines a measure with the required properties.