# L^p (1<p<infty) 的一致凸性

$\displaystyle \|2f\|_{L^p}^p+\|2g\|_{L^p}^p\geq (\|f+g\|_{L^p}+\|f-g\|_{L^p})^p+|\|f+g\|_{L^p}-\|f-g\|_{L^p}|^p.$

$\displaystyle (\|f+g\|_{L^p}+\|f-g\|_{L^p})^p+|\|f+g\|_{L^p}-\|f-g\|_{L^p}|^p\leq 2\cdot 2^p\ \ \ \ \ (1)$

$\displaystyle (a+b)^p+(a-b)^p\geq 2a^p+p(p-1)a^{p-2}b^2\ \ \ \ \ (2).$

(1) $\|f-g\|_{L^p}\leq \|f+g\|_{L^p}$, 根据(1)式和(2)式可得

$\displaystyle 2\|f+g\|_{L^p}^p+p(p-1)\|f+g\|_{L^p}^{p-2}\|f-g\|_{L^p}^2\leq 2\cdot 2^p.$

$\displaystyle \left\|\frac{f+g}{2}\right\|_{L^p}^p\leq 1-\frac{p(p-1)}{2}\left\|\frac{f+g}{2}\right\|_{L^p}^{p-2}\left\|\frac{f-g}{2}\right\|_{L^p}^2\leq 1-\frac{p(p-1)}{2}\left\|\frac{f-g}{2}\right\|_{L^p}^2.$

(2) $\|f+g\|_{L^p}\leq \|f-g\|_{L^p}$，则

$\displaystyle 2\|f-g\|_{L^p}^p+p(p-1)\|f-g\|_{L^p}^{p-2}\|f+g\|_{L^p}^2\leq 2\cdot 2^p.$

$\displaystyle \left\| \frac{f-g}{2} \right\|_{L^p}^p+\frac{p(p-1)}{2}\left\|\frac{f-g}{2}\right\|_{L^p}^{p-2}\left\|\frac{f+g}{2}\right\|_{L^p}^2\leq 1\ \ \ \ \ (3).$

$\|\frac{f+g}{2}\|_{L^p}\leq \frac{1}{2}$时, 我们平凡地有

$\displaystyle \left\|\frac{f+g}{2}\right\|_{L^p}<1-\delta.$

$\displaystyle \left\| \frac{f-g}{2} \right\|_{L^p}^p+\frac{p(p-1)}{2}\left\|\frac{f-g}{2}\right\|_{L^p}^{p-2}\cdot \frac{1}{4}\leq 1.$

$\displaystyle \left\|\frac{f-g}{2}\right\|_{L^p}^p+\frac{1}{8}(p-1)p\leq 1.$

$\displaystyle \left\|\frac{f-g}{2}\right\|_{L^p}<1-\delta.$

# 域扩张

## 代数初步

### 交换环上的代数

$\displaystyle (rx)y=x(ry)=r(xy),\ \ x,y\in A,\ r\in R.$

# 数论问题的价值

(1)第一类问题包括Riemann猜想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。

(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)。

(4)同样，Goldbach猜想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。

【注记】

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# The Kakeya problem in finite fields

The kakeya problem, in its best known formulation, is the following. Let $E\subset\mathbf{R}^n$ be set which contains a translate of every unit segment; equivalently, for every direction $e\in S^{n-1}$, $E$ contains a unit line segment parallel to $e$. An $n$-dimensional ball of radius $1/2$ is a simple example of  a set with this property, but there are many other such sets, some of which have $n$-dimensional measure $0$.

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

# Ordered field

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

### Deformation retract and strong deformation retract

Definition A continuous map

$\displaystyle F:X\times [0,1]\to X$

is a deformation retraction of a space $X$ onto a subspace $A$ if, for every $x$ in $X$ and $a$ in $A$,

$\displaystyle F(x,0)=x,\ F(x,1)\in A,\ \text{and}\ F(a,1)=a.$

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

# Separation theorems in the plane

### 1 The Jordan separation theorem

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 $C$ in the plane separates it into precisely two components, of which $C$ is the common boundary.

Lemma 1 Let $C$ be a compact subspace of $S^2$, let $b$ be a point of $S^2\setminus C$; and let $h$ be a homeomorphism of $S^2\setminus b$ with $\mathbf{R}^2$. Suppose $U$ is a component of $S^2\setminus C$. If $U$ does not contain $b$, then $h(U)$ is a bounded component of $\mathbf{R}^2\setminus h(C)$. If $U$ contains $b$, then $h(U\setminus b)$ is the bounded component of $\mathbf{R}^2\setminus h(C)$.

In particular, if $S^2\setminus C$ has $n$ components, then $\mathbf{R}^2\setminus h(C)$ has $n$ components.

Proof  We show first that if $U$ is a component of $S^2\setminus C$, then $U\setminus \{b\}$ is connencted.

Let $(U_\alpha)$ be the set of components of $S^2\setminus C$; let $V_\alpha=h(U_\alpha\setminus\{b\})$. Because $S^2\setminus C$ is locally connected, the set $U_\alpha$ are connected, disjoint open subsets of $S^2$. Therefore the sets $V_\alpha$ are connected, disjoint, open subsets of $\mathbf{R}^2\setminus h(C)$, so the sets $V_\alpha$ are the components of $\mathbf{R}^2\setminus h(C)$.

Lemma 2 (Nulhomotopy lemma) Let $a$ and $b$ be points of $S^2$. Let $A$ be a compact space, and let

$\displaystyle f:A\to S^2\setminus\{a,b\}$

be a continuous map. If $a$ and $b$ lie in the same component of $S^2\setminus f(A)$, then $f$ is nulhomotopic.

Definition 1 If $X$ is a connected space and $A\subset X$, we say that $A$ separates $X$ if $X\setminus A$ is not conneted; if $X\setminus A$ has $n$ components, we say that $A$ separates $X$ into $n$ components.

Definition 2 An arc is a space homeomorphic to the unit inverval $[0,1]$. The end points of $A$ are the two points $p$ and $q$ such that $A\setminus\{p\}$ and $A\setminus\{q\}$ are connected; the other points of $A$ are called interior points of $A$.

simple closed curve is a space homeomorphic to the unit circle $S^1$.

Theorem 1Suppose $X=U\cup V$, where $U$ and $V$ are open sets of $X$. Suppose that $U\cap V$ is path connected, and that $x_0\in U\cap V$. Let $i$ and $j$ be the inclusion mappings of $U$ and $V$, respectively, into $X$. Then the images of the induced homomorphisms

$\displaystyle i_*:\pi_1(U,x_0)\to \pi_1(X,x_0)\ \ \text{and} \ \ j_*:\pi_1(V,x_0)\to \pi_1(X,x_0)$

generate $\pi_1(X,x_0)$.

Theorem 2 (The Jordan separation theorem) Let $C$ be a simple closed curve in $S^2$. Then $C$ separate $S^2$.

### 2 Invariance of domain

Lemma 3 (Homotopy extension lemma) Let $X$ be a space such that $X\times I$ is normal. Let $A$ be a closed subspace of $X$, and let $f:A\to Y$ be a continuous map, where $Y$ is an open subspace of $\mathbf{R}^n$. If $f$ is nulhomotopic, then $f$ may be extended to a continuous map $g:X\to Y$ that is also nulhomotoptic.

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

Lemma 4 (Borsuk lemma) Let $a$ and $b$ be points of $S^2$. Let $A$ be a compact space, and let $f:A\to S^2\setminus\{a,b\}$ be a continuous injective map. If $f$ is nulhomotopic, then $a$ and $b$ lie in the same component of $S^2\setminus f(A)$.

Theorem 3 (Invariance of domain) If $U$ is an open subset o $\mathbf{R}^2$ and $f:U\to\mathbf{R}^2$ is continuous and injective, then $f(U)$ is open in $\mathbf{R}^2$ and the inverse function $f^{-1}:f(U)\to U$ is continuous.

### 3 The Jordan curve theorem

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 $X=U\cup V$ in the case where the intersection $U\cap V$ is path connected. In the next theorem, we examine what happens when $U\cap V$ 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 $C$ be a simple closed curve in $S^2$. Then $C$ separates $S^2$ into precisely two components $W_1$ and $W_2$. Each of the sets $W_1$ and $W_2$ has $C$ as its boundary; that is, $C=\overline{W}_i-W_i$ for $i=1,2$.

# Application of ultrafilters

An ultrafilter on a set $X$ is a collection $\mathcal{U}$ of subsets of $X$ with the following properties:

1. $\emptyset \notin\mathcal{U}$ and $X\in\mathcal{U}$;
2. $\mathcal{U}$ is closed under finite intersection;
3. if $A\in\mathcal{U}$ and $A\subset B$, then $B\in\mathcal{U}$
4. fr every $A\subset X$, either $A\in\mathcal{U}$ or $X\setminus A\in\mathcal{U}$.

A trivial example of an ultrafilter is the collection of all sets containing some fixed element $x$ of $X$. 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 $\mathcal{U}$ be an ultrafilter on a set $X$.

1. If $X$ is partitioned into finitely many sets $A_1,\dots,A_n$, the precisely one $A_i$ belongs to $\mathcal{U}$.
2. If $F$ and $G$ do not belong to $\mathcal{U}$ then neither does $F\cup G$.
3. If any finite set belongs to $\mathcal{U}$ then $\mathcal{U}$ is a principal filter.

### Examples 1: generalized limits

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

Can we generalize $L$ by finding a linear functional $\phi$ 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 $\phi$ to be linear, and we would like $\phi(a)$ to equal $L(a)$ whenever $a$ is convergent sequence.

If $\mathcal{U}$ is a non-principal ultrafilter, and $(a_1,a_2,\dots)$ is a sequence that takes values in $[-1,1]$, then we can define a limit along $\mathcal{U}$ as follows. Let $\mathcal{J}$ be the collection of all subintervals $J$ of $[-1,1]$ such that $\{n:a_n\in J\}$ belongs $\mathcal{U}$. Then the ultrafilter properties of $\mathcal{U}$ imply that $\mathcal{J}$ has all ultrafilter properties but restricted to intervals.

From this it follows that $\mathcal{J}$ is something like a “principal interval-ultrafilter”. More precisely, it contains all open intervals that contain some particular point $a$. To see this, for each $n\in\mathbf{N}$ partition $[-1,1]$ into finitely many subintervals of length at most $1/n$. Then one of these subintervals belongs to $\mathcal{J}$. So for every $n$ we have an interval $\mathcal{J}_n$ of length $1/n$ that belongs to $\mathcal{J}$. Now let $I_n=J_1\cap \dots\cap J_n$. Since $\mathcal{J}$ is closed under intersection, $I_n$ belongs to $\mathcal{J}$. Let $\{a\}$ be the intersection of the closures of the $I_n$ (which are non-empety and nested). If $U$ is any open interval containing $a$, then $U$ contains some $I_n$, so belongs to $\mathcal{U}$.

Thus ,we have found a number a with the following property: for every $\varepsilon>0$, the set $\{n:|a_n-a|<\varepsilon\}$ belongs to $\mathcal{U}$. Moreover, it is easy to see that this $a$ is unique. We write it as $\lim_{\mathcal{U}}a_n$. It is easy to cheak that $\lim_{\mathcal{U}}$ is linear.

To see ever more clearly how this ties in with the usual notion of a limit, note that $a_n$ converges to $a$ if and only if for every $\varepsilon>0$, the set $\{n:|a_n-a|<\varepsilon\}$ belongs to the cofinite filter.

### Set systems as quantifiers

It is often better to think of a set system as a quantifiers. In particular, if $\mathcal{U}$ is an ultrafilter then one often finds oneself writing sentences of the form $\{x\in X:P(x)\}\in\mathcal{U}$, as we have already seen. But it can be much easier to deal with these sentences if one instead writes $\mathcal{U}x\in X\ P(x)$. One can read this as  “For $\mathcal{U}$-almost every $x\in X\ P(x)$“.

# Lattice property of the class of signed measures

Signed measures have values either in $(-\infty,+\infty]$ or $[-\infty,-\infty)$, to avoid the possibility of adding $+\infty$ to $-\infty$. If $(X,\mathcal{X},\mu)$ is a signed measure space and $A$ is a measurable set, define

$\displaystyle \mu_+(A):=\sup_{B\subset A}\mu(B),\quad \mu_-(A):=-\inf_{B\subset A}\mu(B),\quad |\mu|=\mu_++\mu_-.$

The set function $\mu_-,\mu_+$ and $|\mu|$ are respectively the positive, negative and total variations of $\mu$.

Theorem If $\mu$ and $\nu$ are signed measures on a measurable space, there is a signed measure $\mu\vee \nu$ majorizing $\mu$ and $\nu$ and majorized by every other signed measure majorant $\mu$ and $\nu$.

Proof If $\mu-\nu$ is a well-defined signed measure, that is if $\mu(X)$ and $\nu(X)$ are not both $+\infty$ or both $-\infty$. Let $X_+$ be a maximal positivity set and $X_-$ be a maximal negativity set, for $\mu-\nu$. Define

$\displaystyle (\mu\vee\nu)(A):=\mu(A\cap X_+)+\nu(A\cap X_-).$

This sum defines a measure with the required properties.