Preface

参考摘录于FUNCTIONAL ANALYSIS NOTES——Mr. Andrew Pinchuck

FUNCTIONAL ANALYSIS NOTES

2 Normed Linear Spaces

2.1 Definition

A norm on a linear space XXX is a real-valued function ∥⋅∥:X→R\|\cdot\|: X \rightarrow \mathbb{R}∥⋅∥:X→R which satisfies the following properties:
For all x,y∈Xx, y \in Xx,y∈X and λ∈F\lambda \in \mathbb{F}λ∈F
N1. ∥x∥≥0\|x\| \geq 0∥x∥≥0;
N2. ∥x∥=0⟺x=0\|x\|=0 \Longleftrightarrow x=0∥x∥=0⟺x=0
N3. $\Vert \lambda x\Vert =|\lambda |\Vert x\Vert$
N4. $\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert$ (Triangle Inequality).
A normed linear space is a pair $(X,\Vert \cdot \Vert ),$ where XXX is a linear space and ∥⋅∥\|\cdot\|∥⋅∥ a norm on $X.$ The number $\Vert x\Vert$ is called the norm or length of xxx

Unless there is some danger of confusion, we shall identify the normed linear space $(X,\Vert \cdot \Vert )$ with the underlying linear space XXX.

Examples

[1] Let X=FX=\mathbb{F}X=F. For each x∈Xx \in Xx∈X, define $\Vert x\Vert =|x|$, then (X,∥⋅∥X,\|\cdot\|X,∥⋅∥)is a normed linear spaces

[2] Let nnn be a natural number and $X=\mathbb{F}$. For each x=(x1,x2,…,xn)∈X,x=\left(x_{1}, x_{2}, \ldots, x_{n}\right) \in X,x=(x1​,x2​,…,xn​)∈X, define
∥x∥p=(∑i=1n∣xi∣p)1p,for 1≤p<∞,and ∥x∥∞=max⁡1≤i≤n∣xi∣\begin{array}{l} \|x\|_{p}=\left(\sum_{i=1}^{n}\left|x_{i}\right|^{p}\right)^{\frac{1}{p}}, \text { for } 1 \leq p<\infty, \text { and } \\ \|x\|_{\infty}=\max _{1 \leq i \leq n}\left|x_{i}\right| \end{array} ∥x∥p​=(∑i=1n​∣xi​∣p)p1​,for1≤p<∞,and∥x∥∞​=max1≤i≤n​∣xi​∣​
Then (X,∥⋅∥p)\left(X,\|\cdot\|_{p}\right)(X,∥⋅∥p​) and (X,∥⋅∥∞)\left(X,\|\cdot\|_{\infty}\right)(X,∥⋅∥∞​) are normed linear spaces.

[3] Let $X=\mathcal{B}[a,b]$ be the set of all bounded real-valued functions on $[a,b]$. For each x∈Xx \in Xx∈X, define
∥x∥∞=sup⁡a≤t≤b∣x(t)∣\|x\|_{\infty}=\sup _{a \leq t \leq b}|x(t)| ∥x∥∞​=a≤t≤bsup​∣x(t)∣
Then (X,∥⋅∥∞)\left(X,\|\cdot\|_{\infty}\right)(X,∥⋅∥∞​) is a normed linear space.

[4] Let X=C[a,b]={x:[a,b]→F∣xX=\mathcal{C}[a, b]=\{x:[a, b] \rightarrow \mathbb{F} \mid xX=C[a,b]={x:[a,b]→F∣x is continuous }. For each x∈Xx \in Xx∈X, define
$$
\begin{aligned}

|x|_{\infty} &=\sup _{a \leq t \leq b}|x(t)| \

|x|{2} &=\left(\int{a}^{b}|x(t)|{2} d t\right)^{\frac{1}{2}}

\end{aligned}
$$
Then (X,∥⋅∥∞)\left(X,\|\cdot\|_{\infty}\right)(X,∥⋅∥∞​) and (X,∥⋅∥2)\left(X,\|\cdot\|_{2}\right)(X,∥⋅∥2​) are normed linear spaces.

[5] Let X=ℓp,1≤p<∞X=\ell_{p}, 1 \leq p<\inftyX=ℓp​,1≤p<∞. For each x=(xi)1∞∈X,x=\left(x_{i}\right)_{1}^{\infty} \in X,x=(xi​)1∞​∈X, define
∥x∥p=(∑i∈N∣xi∣p)1p\|x\|_{p}=\left(\sum_{i \in \mathbb{N}}\left|x_{i}\right|^{p}\right)^{\frac{1}{p}} ∥x∥p​=(i∈N∑​∣xi​∣p)p1​
Then (X,∥⋅∥p)\left(X,\|\cdot\|_{p}\right)(X,∥⋅∥p​) is a normed linear space.

[6] Let X=ℓ∞,cX=\ell_{\infty}, cX=ℓ∞​,c or c0.c_{0} .c0​. For each x=(xi)1∞∈X,x=\left(x_{i}\right)_{1}^{\infty} \in X,x=(xi​)1∞​∈X, define
∥x∥=∥x∥∞=sup⁡i∈N∣xi∣\|x\|=\|x\|_{\infty}=\sup _{i \in \mathbb{N}}\left|x_{i}\right| ∥x∥=∥x∥∞​=i∈Nsup​∣xi​∣
Then XXX is a normed linear space.

[7] Let X=L(Cn)X=\mathcal{L}\left(\mathbb{C}^{n}\right)X=L(Cn) be the linear space of all n×nn \times nn×n complex matrices. For A∈L(Cn),A \in \mathcal{L}\left(\mathbb{C}^{n}\right),A∈L(Cn), let τ(A)=∑i=1n(A)ii\tau(A)=\sum_{i=1}^{n}(A)_{i i}τ(A)=∑i=1n​(A)ii​ be the trace of $A.$ For A∈L(Cn),A \in \mathcal{L}\left(\mathbb{C}^{n}\right),A∈L(Cn), define
∥A∥2=τ(A∗A)=∑i=1n∑k=1n(A)ki‾(A)ki=∑i=1n∑k=1n∣(A)ki∣2\|A\|_{2}=\sqrt{\tau\left(A^{*} A\right)}=\sqrt{\sum_{i=1}^{n} \sum_{k=1}^{n} \overline{(A)_{k i}}(A)_{k i}}=\sqrt{\sum_{i=1}^{n} \sum_{k=1}^{n}\left|(A)_{k i}\right|^{2}} ∥A∥2​=τ(A∗A)

​=i=1∑n​k=1∑n​(A)ki​​(A)ki​

​=i=1∑n​k=1∑n​∣(A)ki​∣2

​
where $A$ is the conjugate transpose of the matrix AAA.

Notation

Let aaa be an element of a normed linear space $(X,\Vert \cdot \Vert )$ and $r>0.$

$B(a,r)=\{x\in X\mid \Vert x-a\Vert <r\}$ (Open ball with centre aaa and radius $r)$

$B[a,r]=\{x\in X\mid \Vert x-a\Vert \le r\}$ (Closed ball with centre aaa and radius $r)$

$S(a,r)=\{x\in X\mid \Vert x-a\Vert =r\}$ (Sphere with centre aaa and radius $r)$

Equivalent Norms

Definiton

Let ∥⋅∥\|\cdot\|∥⋅∥ and $\Vert \cdot \Vert$ be two different norms defined on the same linear space $X.$ We say that ∥⋅∥\|\cdot\|∥⋅∥ is equivalent to $\Vert \cdot \Vert$ if there are positive numbers α\alphaα and β\betaβ such that
α∥x∥≤∥x∥0≤β∥x∥,for all x∈X\alpha\|x\| \leq\|x\|_{0} \leq \beta\|x\|, \text { for all } x \in X α∥x∥≤∥x∥0​≤β∥x∥,for allx∈X

example

all norms on a finite-dimensional normed linear space are equivalent.

2.2 Open and Closed Sets

Definition

[1] A subset SSS of a normed linear space $(X,\Vert \cdot \Vert )$ is open if for each s∈Ss \in Ss∈S there is an ϵ>0\epsilon>0ϵ>0 such that B(s,ϵ)⊂SB(s, \epsilon) \subset SB(s,ϵ)⊂S

[2] A subset FFF of a normed linear space $(X,\Vert \cdot \Vert )$ is closed if its complement $X\backslash F$ is open.

[3] Let SSS be a subset of a normed linear space $(X,\Vert \cdot \Vert ).$ We define the closure of $S,$ denoted by $Sˉ,$ to be the intersection of all closed sets containing SSS

It is easy to show that SSS is closed if and only if $S=Sˉ$.

[4] metric on a set XXX is a real-valued function d:X×X→Rd: X \times X \rightarrow \mathbb{R}d:X×X→R which satisfies the following properties: For all x,y,z∈Xx, y, z \in Xx,y,z∈X,
M1. d(x,y)≥0d(x, y) \geq 0d(x,y)≥0
M2. d(x,y)=0⟺x=yd(x, y)=0 \Longleftrightarrow x=yd(x,y)=0⟺x=y
M3. $d(x,y)=d(y,x)$
M4. $d(x,z)\le d(x,y)+d(y,z)$

Theorem

(a) If $(X,\Vert \cdot \Vert )$ is a normed linear space, then
$d(x,y)=\Vert x-y\Vert$
defines a metric on $X.$ Such a metric ddd is said to be induced or generated by the norm $\Vert \cdot \Vert .$ Thus, every normed linear space is a metric space, and unless otherwise specified, we shall henceforth regard any normed linear space as a metric space with respect to the metric induced by its norm.
(b) the property of metric d If ddd is a metric on a linear space XXX satisfying the properties: For all x,y,z∈Xx, y, z \in Xx,y,z∈X and for all λ∈F\lambda \in \mathbb{F}λ∈F,
(i) $d(x,y)=d(x+z,y+z)\text{(Translation Invariance)}$

​ (ii) $d(\lambda x,\lambda y)=|\lambda |d(x,y)$ (Absolute Homogeneity),
then
$\Vert x\Vert =d(x,0)$
defines a norm on XXX.

2.3 Quotient Norm and Quotient Map

[1] Let MMM be a closed linear subspace of a normed linear space XXX over F\mathbb{F}F. The quotient space $X/M$ is a normed linear space with respect to the norm(Quotient Norm)
∥[x]∥:=inf⁡y∈[x]∥y∥=inf⁡m∈M∥x+m∥=inf⁡m∈M∥x−m∥=d(x,M),where [x]∈X/M\|[x]\|:=\inf _{y \in[x]}\|y\|=\inf _{m \in M}\|x+m\|=\inf _{m \in M}\|x-m\|=d(x, M), \text { where }[x] \in X / M ∥[x]∥:=y∈[x]inf​∥y∥=m∈Minf​∥x+m∥=m∈Minf​∥x−m∥=d(x,M),where[x]∈X/M

inf⁡m∈M∥x−m∥=d(x,M)\inf _{m \in M}\|x-m\|=d(x, M)infm∈M​∥x−m∥=d(x,M)，这个我的理解是，x到m的范数最小，也就是x到M的距离，类比点到直线的距离。

[2] Let MMM be a closed subspace of the normed linear space XXX. The mapping $Q$ from $X\to X/M$ defined by
QM(x)=x+M,x∈XQ_{M}(x)=x+M, \quad x \in X QM​(x)=x+M,x∈X
is called the quotient map (or natural embedding) of XXX onto $X/M$.

2.4 Completeness of Normed Linear Spaces

Definition

Let (xn)n=1∞\left(x_{n}\right)_{n=1}^{\infty}(xn​)n=1∞​ be a sequence in a normed linear space $(X,\Vert \cdot \Vert )$.

converge of normed linear space sequence

(a) (xn)n=1∞\left(x_{n}\right)_{n=1}^{\infty}(xn​)n=1∞​ is said to converge to xxx if given ϵ>0\epsilon>0ϵ>0 there exists a natural number $N=N(ϵ)$ such that
∥xn−x∥<ϵfor all n≥N\left\|x_{n}-x\right\|<\epsilon \text { for all } n \geq N ∥xn​−x∥<ϵfor alln≥N
Equivalently, (xn)n=1∞\left(x_{n}\right)_{n=1}^{\infty}(xn​)n=1∞​ converges to xxx if
lim⁡n→∞∥xn−x∥=0\lim _{n \rightarrow \infty}\left\|x_{n}-x\right\|=0 n→∞lim​∥xn​−x∥=0
If this is the case, we shall write
xn→xor lim⁡n→∞xn=xx_{n} \rightarrow x \text { or } \lim _{n \rightarrow \infty} x_{n}=x xn​→xorn→∞lim​xn​=x
Convergence in the norm is called norm convergence or strong convergence.

Cauchy sequence of normed linear space

(b) (xn)n=1∞\left(x_{n}\right)_{n=1}^{\infty}(xn​)n=1∞​ is called a Cauchy sequence if given ϵ>0\epsilon>0ϵ>0 there exists a natural number $N=N(ϵ)$ such that
∥xn−xm∥<ϵfor all n,m≥N\left\|x_{n}-x_{m}\right\|<\epsilon \text { for all } n, m \geq N ∥xn​−xm​∥<ϵfor alln,m≥N
Equivalently, (xn)\left(x_{n}\right)(xn​) is Cauchy if
lim⁡n,m→∞∥xn−xm∥=0\lim _{n, m \rightarrow \infty}\left\|x_{n}-x_{m}\right\|=0 n,m→∞lim​∥xn​−xm​∥=0

Cauchy sequence，在序列号趋于无穷大的时候，它的值就趋于稳定了。

Lemma

1. Let CCC be a closed set in a normed linear space $(X,\Vert \cdot \Vert )$ over $\mathbb{F},$ and let (xn)\left(x_{n}\right)(xn​) be a sequence contained in CCC such that lim⁡n→∞xn=x∈X.\lim _{n \rightarrow \infty} x_{n}=x \in X .limn→∞​xn​=x∈X. Then x∈Cx \in Cx∈C

   赋范空间中的闭合子集中的一个序列，如果收敛，则极限值一定在这个闭合子集中。

2. Let XXX be a normed linear space and AAA a nonempty subset of $X.$
   $[1]|d(x,A)-d(y,A)|\le \Vert x-y\Vert$ for all x,y∈Xx, y \in Xx,y∈X
   $[2]|\Vert x\Vert -\Vert y\Vert |\le \Vert x-y\Vert$ for all x,y∈Xx, y \in Xx,y∈X
   [3] If xn→x,x_{n} \rightarrow x,xn​→x, then ∥xn∥→∥x∥\left\|x_{n}\right\| \rightarrow\|x\|∥xn​∥→∥x∥
   [4] If xn→xx_{n} \rightarrow xxn​→x and yn→y,y_{n} \rightarrow y,yn​→y, then xn+yn→x+yx_{n}+y_{n} \rightarrow x+yxn​+yn​→x+y
   [5] If xn→xx_{n} \rightarrow xxn​→x and αn→α,\alpha_{n} \rightarrow \alpha,αn​→α, then αnxn→αx\alpha_{n} x_{n} \rightarrow \alpha xαn​xn​→αx
   [6] The closure of a linear subspace in XXX is again a linear subspace;
   [7] Every Cauchy sequence is bounded;
   [8] Every convergent sequence is a Cauchy sequence.

Proposition

Let $(X,\Vert \cdot \Vert )$ be a normed linear space over F\mathbb{F}F. A Cauchy sequence in XXX which has a convergent subsequence is convergent.

就是说Cauchy sequence 如果有一个收敛的子序列，那么Cauchy sequence也是收敛的

Completeness

[1] A metric space $(X,d)$ is said to be complete if every Cauchy sequence in XXX converges in XXX.

[2] A normed linear space that is complete with respect to the metric induced by the norm is called a Banach space.

就是说：Banach space是一个metric由norm给出的赋范空间

[3] Theorem Let $(X,\Vert \cdot \Vert )$ be a Banach space and let MMM be a linear subspace of $X.$ Then MMM is complete if and only if the MMM is closed in XXX.

[4] The classical sequence space $\ell$ is complete.

2.5 Series in Normed Linear Spaces

赋范空间中的级数

Definition

[1] Let (xn)\left(x_{n}\right)(xn​) be a sequence in a normed linear space $(X,\Vert \cdot \Vert ).$ To this sequence we associate another sequence (sn)\left(s_{n}\right)(sn​) of partial sums, where sn=∑k=1nxks_{n}=\sum_{k=1}^{n} x_{k}sn​=∑k=1n​xk​

[2] Definition Let (xn)\left(x_{n}\right)(xn​) be a sequence in a normed linear space $(X,\Vert \cdot \Vert ).$ If the sequence (sn)\left(s_{n}\right)(sn​) of partial sums converges to $s,$ then we say that the series ∑k=1∞xk\sum_{k=1}^{\infty} x_{k}∑k=1∞​xk​ converges and that its sum is $s.$ In this case we write ∑k=1∞xk=s\sum_{k=1}^{\infty} x_{k}=s∑k=1∞​xk​=s. The series ∑k=1∞xk\sum_{k=1}^{\infty} x_{k}∑k=1∞​xk​ is said to be absolutely convergent if ∑k=1∞∥xk∥<∞.\sum_{k=1}^{\infty}\left\|x_{k}\right\|<\infty .∑k=1∞​∥xk​∥<∞.

用部分和的形式定义赋范空间中的级数收敛

Theorem

[1] A normed linear space $(X,\Vert \cdot \Vert )$ is a Banach space if and only if every absolutely convergent series in XXX is convergent.

[2] Let MMM be a closed linear subspace of a Banach space $X.$ Then the quotient space $X/M$ is a Banach space when equipped with the quotient norm.

2.6 Bounded, Totally Bounded, and Compact Subsets of a Normed Linear Space

Definition

[1] A subset AAA of a normed linear space $(X,\Vert \cdot \Vert )$ is bounded if $A\subset B[x,r]$ for some x∈Xx \in Xx∈X and r>0r>0r>0
It is clear that AAA is bounded if and only if there is a C>0C>0C>0 such that ∥a∥≤C\|a\| \leq C∥a∥≤C for all a∈Aa \in Aa∈A.

[2] Let AAA be a subset of a normed linear space $(X,\Vert \cdot \Vert )$ and $ϵ>0.$ A subset Aϵ⊂XA_{\epsilon} \subset XAϵ​⊂X is called an ϵ\epsilonϵ -net for AAA if for each x∈Ax \in Ax∈A there is an element $y\in A$ such that $\Vert x-y\Vert <ϵ.$ Simply put, Aϵ⊂XA_{\epsilon} \subset XAϵ​⊂X is an ϵ\epsilonϵ -net for AAA if each element of AAA is within an ϵ\epsilonϵ distance to some element of $A$

$A$表示这样一个集合，对于A中的每一个元素a你总能在$A$中找到对应的某个元素$a$，使得它俩的distance在ϵ\epsilonϵ内。

A subset AAA of a normed linear space $(X,\Vert \cdot \Vert )$ is totally bounded (or precompact) if for any ϵ>0\epsilon>0ϵ>0 there is a finite ϵ\epsilonϵ -net Fϵ⊂XF_{\epsilon} \subset XFϵ​⊂X for AAA. That is, there is a finite set Fϵ⊂XF_{\epsilon} \subset XFϵ​⊂X such that
A⊂⋃x∈FϵB(x,ϵ)A \subset \bigcup_{x \in F_{\epsilon}} B(x, \epsilon) A⊂x∈Fϵ​⋃​B(x,ϵ)

Proposition

[1] Every totally bounded subset of a normed linear space $(X,\Vert \cdot \Vert )$ is bounded.

[2] A subset AAA of a normed linear space $(X,\Vert \cdot \Vert )$ is totally bounded if and only if for any ϵ>0\epsilon>0ϵ>0 there is a finite set Fϵ⊂AF_{\epsilon} \subset AFϵ​⊂A such that
A⊂⋃x∈FϵB(x,ϵ)A \subset \bigcup_{x \in F_{\epsilon}} B(x, \epsilon) A⊂x∈Fϵ​⋃​B(x,ϵ)
[3] A normed linear space $(X,\Vert \cdot \Vert )$ is sequentially compact if every sequence in XXX has a convergent subsequence.

Theorem

[1] A subset KKK of a normed linear space $(X,\Vert \cdot \Vert )$ is totally bounded if and only if every sequence in KKK has a Cauchy subsequence.

[2] A subset of a normed linear space is sequentially compact if and only if it is totally bounded and complete .

Remark

It can be shown that on a metric space, compactness and sequential compactness are equivalent. Thus, it follows, that on a normed linear space, we can use these terms interchangeably.

Corollary

[1] A subset of a Banach space is sequentially compact if and only if it is totally bounded and closed

[2] A sequentially compact subset of a normed linear space is closed and bounded.

[3] A closed subset F of a sequentially compact normed linear space $（X;\Vert \cdot \Vert ）$is sequentially compact.

泛函分析2——Normed Linear Spaces相关推荐

1. 双语矩阵论课程笔记（2）—— 【chapter 1】 Vector Spaces (Linear Spaces)

   双语矩阵论课程笔记 文章目录 1. Definitions and Examples 1.1 Number filed(数域) 1.2 Algebraic systems(代数系统) 1.3 Line ...

2. 泛函分析——有界线性算子和函数

   线性映射: Let XXX and YYY be linear spaces over the same field F\mathbb{F}F. A linear operator from XXX ...

3. 泛函分析4——希尔伯特空间

   文章目录 它只是为了凑个标题 3 Hilbert Spaces 3.1Introduction Definition Example Theorem Corollary 3.2 Hilbert Spa ...

4. 泛函分析——内积空间定义的概念

   希尔伯特空间: Let (X,⟨⋅,⋅⟩)(X,\langle\cdot, \cdot\rangle)(X,⟨⋅,⋅⟩) be an inner product space. If XXX is co ...

5. Introduction to the calculus of variations

   转自:http://staff.www.ltu.se/~larserik/applmath/chap3en/ Parts: 1. Functions - extreme points. 2. Func ...

6. 泛函分析——赋范线性空间定义的概念

   赋范线性空间定义的概念 开集: A subset SSS of a normed linear space (X,∥⋅∥)(X,\|\cdot\|)(X,∥⋅∥) is open if for eac ...

7. 泛函分析1-线性空间

   文章目录 Preface 1 Linear Spaces 1.1 linear space Examples 1.2 Subsets of a linear space Notation 1.3 Su ...

8. 概念和术语-数学统计学

   目录 其他 参数模型和非参数模型 曼哈顿距离 解析解/闭式解 先验概率和后验概率 似然函数 凸集.凸函数和凸优化 histogram bin Jaccard指数 协方差 协方差矩阵 双曲函数 插值 & ...

9. 数学基础 - 线性空间（Vector Space）

   线性空间(Vector Space) 定义(Definition) 设V是一个非空集合,P是一个域: 加法: ∀α,β∈V ∀ α , β ∈ V \forall \; \alpha,\beta \i ...

最新文章

1. python电脑安装-如何安装python
2. Swift之深入解析构造过程和析构过程
3. PHP 获取IP地址 获取当前时间 写文件
4. Qt网络程序：基于TCP的服务器、客户端实例
5. 使用电脑adb给Essential Phone刷机 —(官方篇)
6. Go中的三种排序方法
7. 住个酒店，“我”的隐私就被强制扒干净了！
8. 一个关于如何创建类似于QQ客户端聊天窗口的模拟小项目
9. 大白话解析模拟退火算法、遗传算法
10. springmvc配置拦截器及测试详解
11. 110报警声+单片机c语言,单片机实现消防车报警声音的设计
12. 【CVPR-2019】基于深度学习优化光照的暗光图像增强
13. nssl 1467.U
14. [转载]网络数据流的java处理
15. oracle stdevp函数,SQL Server和Oracle的常用函数对比
16. S3C2440 音频解码芯片WM8976声卡驱动移植、测试以及madplay播放mp3文件（三十一）
17. 北理工计算机学院沈建斌,沈建冰_北京理工大学计算机学院
18. 04-再探JavaScript
19. jdk版本特性一览（持续补充新版本）
20. 知乎：为什么我的成绩那么好，最终还是成了一个没用的人

热门文章

1. 【技术干货】浅析State-Thread
2. 请回答! 2018泛娱乐风口
3. 比特币的超能力：永恒不变
4. 云栖科技评论第53期：前沿技术要“会说话”
5. php程序中报date错误
6. 轻量集群管理工具PSSH
7. HDU 1875 畅通工程再续 (最小生成树)
8. 【hibernate框架】一对多(多对一)双向CRUD-关于delete2
9. SQL Server远程部署
10. 利用公众号 + WeRoBot开发一款AI应用