# 数学代写|拓扑学代写Topology代考|ORTHONORMAL SETS

## 数学代写|拓扑学代写Topology代考|ORTHONORMAL SETS

An orthonormal set in a Hilbert space $H$ is a non-empty subset of $H$ which consists of mutually orthogonal unit vectors; that is, it is a nonempty subset $\left{e_i\right}$ of $H$ with the following properties:
(1) $i \neq j \Rightarrow e_i \perp e_j$
(2) $\left|e_i\right|=1$ for every $i$.
If $H$ contains only the zero vector, then it has no orthonormal sets. If $H$ contains a non-zero vector $x$, and if we normalize $x$ by considering $e=x /|x|$, then the single-element set ${e}$ is clearly an orthonormal set. More generally, if $\left{x_i\right}$ is a non-empty set of mutually orthogonal non-zero vectors in $H$, and if the $x_i$ ‘s are normalized by replacing each of them by $e_i=x_i /\left|x_i\right|$, then the resulting set $\left{e_i\right}$ is an orthonormal set.

Example 1. The subset $\left{e_1, e_2, \ldots, e_n\right}$ of $l_2^n$, where $e_i$ is the $n$-tuple with 1 in the $i$ th place and 0 ‘s elsewhere, is evidently an orthonormal set in this space.

Example 2. Similarly, if $e_n$ is the sequence with 1 in the $n$th place and 0 ‘s elsewhere, then $\left{e_1, e_2, \ldots, e_n, \ldots\right}$ is an orthonormal set in $l_2$.
At the end of this section, we give some additional examples taken from the field of analysis.

Every aspect of the theory of orthonormal sets depends in one way or another on our first theorem.

Theorem A. Let $\left{e_1, e_2, \ldots, e_n\right}$ be a finite orthonormal set in a Hilbert space $H$. If $x$ is any vector in $H$, then
\begin{aligned} & \sum_{i=1}^n\left|\left(x, e_i\right)\right|^2 \leq|x|^2 \ & x-\sum_{i=1}^n\left(x, e_i\right) e_i \perp e_j \end{aligned}
further,
for each $j$.

## 数学代写|拓扑学代写Topology代考|THE CONJUGATE SPACE H

We pointed out in the introduction to this chapter that one of the fundamental properties of a Hilbert space $H$ is the fact that there is a natural correspondence between the vectors in $H$ and the functionals in $H^*$. Our purpose in this section is to develop the features of this correspondence which are relevant to our work with operators in the rest of the chapter.

Let $y$ be a fixed vector in $H$, and consider the function $f_v$ defined on $H$ by $f_y(x)=(x, y)$. It is easy to see that $f_y$ is linear, for
and
\begin{aligned} f_v\left(x_1+x_2\right) & =\left(x_1+x_2, y\right) \ & =\left(x_1, y\right)+\left(x_2, y\right) \ & =f_v\left(x_1\right)+f_y\left(x_2\right) \ f_y(\alpha x) & =(\alpha x, y) \ & =\alpha(x, y) \ & =\alpha f_y(x) . \end{aligned}

Further, $f_y$ is continuous and is therefore a functional, for Schwarz’s inequality gives
\begin{aligned} \left|f_y(x)\right| & =|(x, y)| \ & \leq|x||y|, \end{aligned}
which shows that $\left|f_y\right| \leq|y|$. Even more, equality is attained here, that is, $\left|f_y\right|=|y|$. This is clear if $y=0$; and if $y \neq 0$, it follows from
\begin{aligned} \left|f_y\right| & =\sup \left{\left|f_y(x)\right|:|x|=1\right} \ & \geq\left|f_v\left(\frac{y}{|y|}\right)\right| \ & =\left|\left(\frac{y}{|y|}, y\right)\right|=|y| . \end{aligned}

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|ORTHONORMAL SETS

(1) $i \neq j \Rightarrow e_i \perp e_j$
(2) $\left|e_i\right|=1$ 每一个 $i$.

$\backslash$ \eft{e_1, e_2, \dots, e_n, \Idots \right } } \text { 是一个正交集 } l _ { 2 } \text { . }

$$\sum_{i=1}^n\left|\left(x, e_i\right)\right|^2 \leq|x|^2 \quad x-\sum_{i=1}^n\left(x, e_i\right) e_i \perp e_j$$

## 数学代写|拓扑学代写Topology代考|THE CONJUGATE SPACE H

$$f_v\left(x_1+x_2\right)=\left(x_1+x_2, y\right) \quad=\left(x_1, y\right)+\left(x_2, y\right)=f_v\left(x_1\right)+f_y\left(x_2\right) f_y(\alpha x)$$

$$\left|f_y(x)\right|=|(x, y)| \quad \leq|x||y|,$$

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: