### 数学代写|随机过程作业代写Stochastic Processes代考|Basic notions in functional analysis

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• Statistical Inference 统计推断
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|随机过程作业代写Stochastic Processes代考|Linear spaces

The central notion of functional analysis is that of a Banach space. There are two components of this notion: algebraic and topological. The algebraic component describes the fact that elements of a Banach space may be meaningfully added together and multiplied by scalars. For example, given two random variables, $X$ and $Y$, say, we may think of random variables $X+Y$ and $\alpha X$ (and $\alpha Y$ ) where $\alpha \in \mathbb{R}$. In a similar way, we may think of the sum of two measures and the product of a scalar and a measure. Abstract sets with such algebraic structure, introduced in more detail in this section, are known as linear spaces. The topological component of the notion of a Banach space will be discussed in Section $2.2$.

2.1.1 Definition Let $X$ be a set; its elements will be denoted $x, y, z$, etc. A triple $(\mathbb{X},+, \cdot)$, where $+$ is a map $+: \mathbb{X} \times \mathbb{X} \rightarrow \mathbb{X},(x, y) \mapsto x+y$ and – is a map $:: \mathbb{R} \times \mathbb{X} \rightarrow \mathbb{X},(\alpha, x) \mapsto \alpha x$, is called a (real) linear space if the following conditions are satisfied:
(a1) $(x+y)+z=x+(y+z)$, for all $x, y, z \in \mathbb{X}$,
(a2) there exists $\Theta \in \mathbb{X}$ such that $x+\Theta=x$, for all $x \in \mathbb{X}$,
(a3) for all $x \in \mathbb{X}$ there exists an $x^{\prime} \in \mathbb{X}$ such that $x+x^{\prime}=\Theta$,
(a4) $x+y=y+x$, for all $x, y \in \mathbb{X}$,
(m1) $\alpha(\beta x)=(\alpha \beta) x$, for all $\alpha, \beta \in \mathbb{R}, x \in \mathbb{X}$,
(m2) $1 x=x$, for all $x \in \mathbb{X}$,
(d) $\alpha(x+y)=\alpha x+\alpha y$, and $(\alpha+\beta) x=\alpha x+\beta x$ for all $\alpha, \beta \in \mathbb{R}$ and $x, y \in \mathbb{X}$.

Conditions (a1)-(a4) mean that $(\mathbb{X},+)$ is a commutative group. Quite often, for the sake of simplicity, when no confusion ensues, we will say that $\mathrm{X}$ itself is a linear space.
2.1.2 Exercise Conditions (a2) and (a4) imply that the element $\Theta$, called the zero vector, or the zero, is unique.
2.1.3 Exercise Conditions (a1) and (a3) -(a4) imply that for any $x \in$ $X, x^{\prime}$ is determined uniquely.
2.1.4 Exercise Conditions (d), (a1) and (a3) imply that for any $x \in$ $\mathrm{X}, 0 x=\Theta$.
2.1.5 Exercise 2.1.3, and 2.1.1 (d), (m2) imply that for any $x \in \mathbb{X}$, $x^{\prime}=(-1) x$. Because of this fact, we will adopt the commonly used notation $x^{\prime}=-x$.
2.1.6 Example Let $S$ be a set. The set $\mathbb{X}=\mathbb{R}^{S}$ of real-valued functions defined on $S$ is a linear space, if addition and multiplication are defined as follows: $(x+y)(p)=x(p)+y(p),(\alpha x)(p)=\alpha x(p)$, for all $x(\cdot), y(\cdot) \in \mathbb{R}^{S}, \alpha \in \mathbb{R}$, and $p \in S$. In particular, the zero vector $\Theta$ is a function $x(p) \equiv 0$, and $-x$ is defined by $(-x)(p) \equiv-x(p)$. This example includes a number of interesting subcases: (a) if $S=\mathbb{N}, \mathbb{R}^{\mathbb{N}}$ is the space of real-valued sequences, (b) if $S=\mathbb{R}, \mathbb{R}^{\mathbb{R}}$ is the space of real functions on $\mathbb{R}$, (c) if $S={1, \ldots, n} \times{1,2, \ldots, k}, \mathbb{R}^{S}$ is the space of real $n \times k$ matrices, etc.

## 数学代写|随机过程作业代写Stochastic Processes代考|Banach spaces

As we have mentioned already, the notion of a Banach space is crucial in functional analysis and in this book. Having covered the algebraic aspects of Banach space in the previous section, we now turn to discussing topological aspects. A natural way of introducing topology in a linear space is by defining a norm. Hence, we begin this section with the definition of a normed space (which is a linear space with a norm) and continue with discussion of Cauchy sequences that leads to the definition of a Banach space, as a normed space “without holes”. Next, we give a number of examples of Banach spaces (mostly those that are important in probability theory) and introduce the notion of isomorphic Banach spaces. Then we show how to immerse a normed space in a Banach space and provide examples of dense algebraic subspaces of Banach spaces. We close by showing how the completeness of a Banach space may be used to prove existence of an element that satisfies some required property.

2.2.1 Normed linear spaces Let $X$ be a linear space. A function $|\cdot|$ : $\mathrm{X} \rightarrow \mathbb{R}, x \mapsto|x|$ is called a norm, if for all $x, y \in \mathbb{X}$ and $\alpha \in \mathbb{R}$,
(n1) $|x| \geq 0$,
(n2) $|x|=0$, iff $x=\Theta$,
(n3) $|\alpha x|=|\alpha||x|$,
(n4) $|x+y| \leq|x|+|y|$.
If (n2) does not necessarily hold, $|\cdot|$ is called a semi-norm. Note that if $|\cdot|$ is a semi-norm, then $|\Theta|=0$ by (n3) and 2.1.4. A pair (X, $|\cdot|)$, where $X$ is a linear space and $|\cdot|$ is a norm in $X$ called a normed linear space, and for simplicity we say that $X$ itself is a normed linear space (or just normed space).
2.2.2 Exercise $(n 3)-(n 4)$ imply that for $x, y \in \mathbb{X}$,
$$||x|-|y|| \leq|x \pm y| .$$

## 数学代写|随机过程作业代写Stochastic Processes代考|The space of bounded linear operators

Throughout this section, $\left(\mathbb{X},|\cdot|_{X}\right)$ and $\left(\mathbb{Y},|\cdot|_{Y}\right)$ are two linear normed spaces. From now on, to simplify notation, we will denote the zero vector in both spaces by 0 .
2.3.1 Definition A linear map $L: \mathbb{X} \rightarrow \mathbb{Y}$ is said to be bounded if $|L x|_{\mathrm{Y}} \leq M|x|_{\mathrm{X}}$ for some $M \geq 0$. If $M$ can be chosen equal to $1, L$ is called a contraction. In particular, isometric isomorphisms are contractions. Linear contractions, i.e. linear operators that are contractions are very important for the theory of stochastic processes, and appear often.
2.3.2 Definition As in 2.1.12, we show that the collection $\mathcal{L}(\mathbb{X}, \mathbb{Y})$ of continuous linear operators from $\mathbb{X}$ to $\mathbb{Y}$ is an algebraic subspace of $L(\mathbb{X}, \mathbb{Y}) . \mathcal{L}(\mathbb{X}, \mathbb{Y})$ is called the space of bounded (or continuous) linear operators on $X$ with values in $Y$. The first of these names is justified by the fact that a linear operator is bounded iff it is continuous, as proved below. If $\mathbb{X}=\mathbb{Y}$ we write $\mathcal{L}(\mathbb{X})$ instead of $\mathcal{L}(\mathbb{X}, \mathbb{Y})$ and call this space the space of bounded linear operators on $\mathbb{X}$. If $\mathbb{Y}=\mathbb{R}$, we write $\mathbb{X}^{*}$ instead of $\mathcal{L}(\mathbb{X}, \mathbb{Y})$ and call it the space of bounded linear functionals on $\mathrm{X}$.

2.3.3 Theorem Let $L$ belong to $L(\mathbb{X}, Y)$ (see 2.1.7). The following conditions are equivalent:
(a) $L$ is continuous $(L \in \mathcal{L}(\mathbb{X}, Y))$,
(b) $L$ is continuous at some $x \in X$,
(c) $L$ is continuous at zero,
(d) $\sup {|x| x=1}|L x|{\mathrm{y}}$ is finite,
(e) $L$ is bounded.
Moreover, $\sup {|x|{x}=1}|L x|_{Y}=\min {M \in \mathcal{M}}$ where $\mathcal{M}$ is the set of constants such that $|L x|_{\mathrm{Y}} \leq M|x|_{\mathrm{X}}$ holds for all $x \in \mathrm{X}$.

Proof The implication $(a) \Rightarrow(b)$ is trivial. If a sequence $x_{n}$ converges to zero, then $x_{n}+x$ converges to $x$. Thus, if (b) holds, then $L\left(x_{n}+x\right)$, which equals $L x_{n}+L x$, converges to $L x$, i.e. $L x_{n}$ converges to 0 , showing (c). To prove that (c) implies (d), assume that (d) does not hold, i.e. there exists a sequence $x_{n}$ of elements of $\mathrm{X}$ such that $\left|x_{n}\right| \mathrm{X}=1$ and $\left|L x_{n}\right|>n$. Then the sequence $y_{n}=\frac{1}{\sqrt{n}} x_{n}$ converges to zero, but $\left|L y_{n}\right|_{Y}>\sqrt{n}$ must not converge to zero, so that (c) does not hold. That (d) implies (e) is seen by putting $M=\sup {|x|{\mathrm{x}}}|L x|_{\mathrm{Y}}$; indeed, the inequality in the definition 2.3.1 is trivial for $x=0$, and for a non-zero vector $x$, the norm of $\frac{1}{|x|} x$ equals one, so that $\left|L \frac{1}{|x|} x\right|_{\mathrm{y}} \leq M$, from which (e) follows by multiplying both sides by $|x|$. Finally, (a) follows from (e), since $\left|L x_{n}-L x\right| \leq\left|L\left(x_{n}-x\right)\right| \leq M\left|x_{n}-x\right| .$

To prove the second part of the theorem, note that in the proof of the implication $(\mathrm{d}) \Rightarrow(\mathrm{e})$ we showed that $M_{1}=\sup {|x| \mathrm{x}=1}|L x|{\mathrm{Y}}$ belongs to $\mathcal{M}$. On the other hand, if $|L x|_{\mathrm{Y}} \leq M|x|_{\mathrm{X}}$ holds for all $x \in X$, then considering only $x$ with $|x|_{\mathrm{x}}=1$ we see that $M_{1} \leq M$ so that $M_{1}$ is the minimum of $\mathcal{M}$.

## 数学代写|随机过程作业代写Stochastic Processes代考|Linear spaces

2.1.1 定义让X成为一个集合；它的元素将被表示X,是,和等。(X,+,⋅)， 在哪里+是一张地图+:X×X→X,(X,是)↦X+是和 – 是一张地图::R×X→X,(一种,X)↦一种X，如果满足以下条件，则称为（实）线性空间：
（a1）(X+是)+和=X+(是+和)， 对全部X,是,和∈X,
(a2) 存在θ∈X这样X+θ=X， 对全部X∈X,
(a3) 对于所有X∈X存在一个X′∈X这样X+X′=θ,
(a4)X+是=是+X， 对全部X,是∈X,
(m1)一种(bX)=(一种b)X， 对全部一种,b∈R,X∈X,
(m2)1X=X， 对全部X∈X,
(d)一种(X+是)=一种X+一种是， 和(一种+b)X=一种X+bX对全部一种,b∈R和X,是∈X.

2.1.2 运动条件 (a2) 和 (a4) 暗示元素θ，称为零向量或零，是唯一的。
2.1.3 行使条件 (a1) 和 (a3) -(a4) 意味着对于任何X∈ X,X′是唯一确定的。
2.1.4 行使条件 (d)、(a1) 和 (a3) 意味着对于任何X∈ X,0X=θ.
2.1.5 练习 2.1.3 和 2.1.1 (d), (m2) 暗示对于任何X∈X, X′=(−1)X. 由于这个事实，我们将采用常用的符号X′=−X.
2.1.6 例子让小号成为一个集合。套装X=R小号上定义的实值函数小号是一个线性空间，如果加法和乘法定义如下：(X+是)(p)=X(p)+是(p),(一种X)(p)=一种X(p)， 对全部X(⋅),是(⋅)∈R小号,一种∈R， 和p∈小号. 特别是，零向量θ是一个函数X(p)≡0， 和−X定义为(−X)(p)≡−X(p). 这个例子包括一些有趣的子案例：（a）如果小号=ñ,Rñ是实值序列的空间，(b) 如果小号=R,RR是实函数的空间R, (c) 如果小号=1,…,n×1,2,…,ķ,R小号是真实的空间n×ķ矩阵等

## 数学代写|随机过程作业代写Stochastic Processes代考|Banach spaces

2.2.1 范数线性空间 LetX是一个线性空间。一个函数|⋅| : X→R,X↦|X|被称为规范，如果对所有人X,是∈X和一种∈R,
(n1)|X|≥0,
(n2)|X|=0, 当且X=θ,
(n3)|一种X|=|一种||X|,
(n4)|X+是|≤|X|+|是|.

2.2.2 练习(n3)−(n4)暗示对于X,是∈X,
||X|−|是||≤|X±是|.

## 数学代写|随机过程作业代写Stochastic Processes代考|The space of bounded linear operators

2.3.1 定义线性映射大号:X→是据说是有界的，如果|大号X|是≤米|X|X对于一些米≥0. 如果米可以选择等于1,大号称为收缩。特别是，等距同构是收缩。线性收缩，即作为收缩的线性算子对于随机过程的理论非常重要，并且经常出现。
2.3.2 定义与 2.1.12 一样，我们展示了集合大号(X,是)来自的连续线性算子X到是是一个代数子空间大号(X,是).大号(X,是)称为有界（或连续）线性算子的空间X与值是. 这些名称中的第一个是由以下事实证明的：线性算子是有界的，如果它是连续的，如下所示。如果X=是我们写大号(X)代替大号(X,是)并将这个空间称为有界线性算子的空间X. 如果是=R， 我们写X∗代替大号(X,是)并将其称为有界线性泛函空间X.

2.3.3 定理让大号属于大号(X,是)（见 2.1.7）。以下条件是等效的：
(a)大号是连续的(大号∈大号(X,是)),
(b)大号在某些地方是连续的X∈X,
(c)大号在零处连续，
(d) $\sup {|x| x=1}|长 x| {\mathrm{y}}一世sF一世n一世吨和,(和)大号一世sb这在nd和d.米这r和这在和r,\sup {|x| {x}=1}|L x|_{Y}=\min {M \in \mathcal{M}}在H和r和\数学{M}一世s吨H和s和吨这FC这ns吨一种n吨ss在CH吨H一种吨|L x|_{\mathrm{Y}} \leq M|x|_{\mathrm{X}}H这ldsF这r一种llx \in \mathrm{X}$。

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