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# 数学代写|拓扑学代写TOPOLOGY代考|MATH6280 ORTHOGONAL COMPLEMENTS

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## 数学代写|拓扑学代写TOPOLOGY代考|ORTHOGONAL COMPLEMENTS

Two vectors $x$ and $y$ in a Hilbert space $H$ are said to be orthogonal (written $x \perp y$ ) if $(x, y)=0$. The symbol $\perp$ is of ten pronounced “perp.” Since $\overline{(x, y)}=(y, x)$, we have $x \perp y \Leftrightarrow y \perp x$. It is also clear that $x \perp 0$ for every $x$, and $(x, x)=|x|^2$ shows that 0 is the only vector orthogonal to itself. One of the simplest geometric facts about orthogonal vectors is the Pythagorean theorem:
$$x \perp y \Rightarrow|x+y|^2=|x-y|^2=|x|^2+|y|^2 .$$
A vector $x$ is said to be orthogonal to a non-empty set $S$ (written $x \perp S$ ) if $x \perp y$ for every $y$ in $S$, and the orthogonal complement of $S$-denoted by $S^{\perp}$-is the set of all vectors orthogonal to $S$. The following statements are easy consequences of the definition:
$$\begin{gathered} {0} \perp=H ; H \perp={0} ; \ S \cap H^{\perp} \subseteq{0} ; \ S_1 \subseteq S_2 \Rightarrow S_1 \perp \supseteq S_2{ }^{\perp} \end{gathered}$$
$S^{\perp}$ is a closed linear subspace of $H$.

## 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.

## 数学代写|拓扑学代写TOPOLOGY代考|ORTHOGONAL COMPLEMENTS

$$x \perp y \Rightarrow|x+y|^2=|x-y|^2=|x|^2+|y|^2 .$$

$$0 \perp=H ; H \perp=0 ; S \cap H^{\perp} \subseteq 0 ; S_1 \subseteq S_2 \Rightarrow S_1 \perp \supseteq S_2{ }^{\perp}$$
$S^{\perp}$ 是一个封闭的线性子空间 $H$.

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

《left 缺少或无法识别的分隔符

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

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