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# 数学代写|泛函分析作业代写functional analysis代考|Hilbert Spaces and Their Operators

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## 数学代写|泛函分析作业代写functional analysis代考|Definition, Examples and Basic Properties

From now on, $\mathbf{F}$ still denotes the field of real or complex numbers.
Definition 10.1.1. Let $X$ be a linear space over $\mathbf{F}$. An inner product on $X$ is $a$ $\operatorname{map}(x, y) \mapsto\langle x, y\rangle$ from $X \times X$ to $\mathbf{F}$ such that:
(a) $\langle\alpha x+\beta y, z\rangle=\alpha\langle x, z\rangle+\beta\langle y, z\rangle$ for all $x, y, z \in X$ and for all $\alpha, \beta \in \mathbf{F}$;
(b) $\langle y, x\rangle=\overline{\langle x, y\rangle}$ for all $x, y \in X$;
(c) $\langle x, x\rangle \geq 0$ for all $x \in X$ and $\langle x, x\rangle=0$ if and only if $x=0$.
If $X$ is equipped with $\langle\cdot, \cdot\rangle$, then it is called an inner product space or a preHilbert space over $\mathbf{F}$.

Note that the bar signifies complex conjugation – of course, if the scalar field is $\mathbf{R}$ then the axiom (b) is simply $\langle y, x\rangle=\langle x, y\rangle$. So we will refer to real or complex Hilbert spaces depending on whether $\mathbf{F}$ is $\mathbf{R}$ or $\mathbf{C}$.

## 数学代写|泛函分析作业代写functional analysis代考|Orthogonality, Orthogonal Complement and Duality

First of all, let us consider orthogonality.
Definition 10.2.1. Let $X$ be a Hilbert space over $\mathbf{F}$.
(i) The angle between two vectors $x, y \in X$ is defined by
$$\theta_{x, y}=\left{\begin{array}{l} 0, \text { x or } y=0 \ \arccos \frac{\Re\langle x, y\rangle}{|x| y||}, \text { otherwise } \end{array} .\right.$$
(ii) Two vectors $x, y \in X$ are called orthogonal, denoted $x \perp y$, provided $\langle x, y\rangle=0$.
(iii) For any subset $S$ of $X$, the set $S^{\perp}={x \in X:\langle x, y\rangle=0$ for all $y \in S}$ is called the orthogonal complement of $S$.

## 数学代写|泛函分析作业代写FUNCTIONAL ANALYSIS代考|Orthonormal Sets and Bases

To better understand the structure of a Hilbert space, in this section we consider the concepts of orthonormal sets and bases.
Hilbert Spaces and Their Operators
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Definition 10.3.1. Let $X$ be a Hilbert space over $\mathbf{F}$. A set $S \subseteq X$ is called orthogonal if any two different elements in $S$ are orthogonal. An orthonormal set is an orthogonal set consisting entirely of elements of norm $1 .$

A constructive method of orthonormalizing a set of vectors in a Hilbert space is the Gram-Schmidt process, named for Jørgen Pedersen Gram and Erhard Schmidt, as follows.

Theorem 10.3.2. Let $\left{x_{j}\right}_{j=1}^{\infty}$ be a countable linearly independent set of a Hilbert space $X$ over $\mathbf{F}$. Then a countable orthonormal set $\left{e_{j}\right}$ can be constructed so that
$$\operatorname{span}\left{e_{j}\right}_{j=1}^{n}=\operatorname{span}\left{x_{j}\right}_{j=1}^{n} \text { for all } n \in \mathbf{N}$$

## 数学代写|泛函分析作业代写FUNCTIONAL ANALYSIS代考|DEFINITION, EXAMPLES AND BASIC PROPERTIES

b ⟨是,X⟩=⟨X,是⟩¯对全部X,是∈X;
C ⟨X,X⟩≥0对全部X∈X和⟨X,X⟩=0当且仅当X=0.

## 数学代写|泛函分析作业代写FUNCTIONAL ANALYSIS代考|ORTHOGONALITY, ORTHOGONAL COMPLEMENT AND DUALITY

$$\theta_{x, y}=\left{ 定义0, x 或 是=0 阿尔科斯⁡ℜ⟨X,是⟩|X|是||, 除此以外 。\对。$$

## 数学代写|泛函分析作业代写FUNCTIONAL ANALYSIS代考|ORTHONORMAL SETS AND BASES

305

\$$\operatorname{span}\left{e_{j}\right}_{j=1}^{n}=\operatorname{span}\left{x_{j}\right}_{j=1}^{n} \text { for all } n \in \mathbf{N}$$