# 数学代写|泛函分析作业代写Functional Analysis代考|MATH4101

## 数学代写|泛函分析作业代写Functional Analysis代考|Banach Lattices

Over the real scalar field, all Banach spaces discussed in this chapter are examples of Banach lattices, a class of Banach spaces that will be briefly discussed in this section.

The main result, Theorem $2.57$, shows that any complete norm on a Banach lattice $X$ which is monotone with respect to the partial order of $X$ is equivalent to the given norm of $X$.

Let $(S, \leqslant)$ be a partially ordered set and let $S^{\prime}$ be a subset of $S$. An element $x \in S$ is said to be a lower bound for $S^{\prime}$ if we have $x \leqslant x^{\prime}$ for all $x^{\prime} \in S^{\prime}$. Such an element is called a greatest lower bound for $S^{\prime}$ if $y \leqslant x$ holds for every lower bound $y$ for $S^{\prime}$. Similarly an element $x \in S$ is said to be an upper bound for $S^{\prime}$ if we have $x^{\prime} \leqslant x$ for all $x^{\prime} \in S^{\prime}$, and such an element is called a least upper bound for $S^{\prime}$ if $x \leqslant y$ holds for every upper bound $y$ for $S^{\prime}$. Greatest lower bounds and least upper bounds, if they exist, are unique.

Definition 2.50 (Lattices). A partially ordered set $(S, \leqslant)$ is called a lattice if every pair of elements has a greatest lower bound and a least upper bound.

The greatest lower bound and the least upper bound of the pair ${x, y} \subseteq S$ in a partially ordered set $S$ will be denoted by $x \wedge y$ and $x \vee y$, respectively.

## 数学代写|泛函分析作业代写Functional Analysis代考|Hilbert Spaces

Let $V$ be a vector space. A mapping $\phi: V \times V \rightarrow \mathbb{K}$ is called sesquilinear if it is linear in the first variable and conjugate-linear in the second variable, that is,
$\phi\left(v+v^{\prime}, w\right)=\phi(v, w)+\phi\left(v^{\prime}, w\right), \quad \phi(c v, w)=c \phi(v, w)$,
$\phi\left(v, w+w^{\prime}\right)=\phi(v, w)+\phi\left(v, w^{\prime}\right), \quad \phi(v, c w)=\bar{c} \phi(v, w)$,
for all $c \in \mathbb{K}$ and $v, v^{\prime}, w, w^{\prime} \in V$. The complex conjugation in (3.2) is of course redundant when the scalar field is real and sesquilinearity reduces to bilinearity in that case.

Definition 3.1 (Inner products). An inner product space is a pair $(H,(\cdot \mid \cdot))$, where $H$ is a vector space and $(\cdot \mid \cdot)$ is an inner product on $H \times H$, that is, a sesquilinear mapping from $H \times H$ to $\mathbb{K}$ with the following properties:
(i) $(x \mid x) \geqslant 0$ for all $x \in H$ and $(x \mid x)=0 \Rightarrow x=0$;
(ii) $(x \mid y)=\overline{(y \mid x)}$ for all $x, y \in H$.
The conjugation bar in (ii) is again redundant when the scalar field is real. If (ii) holds, then (3.1) implies (3.2).

It will be used frequently without further comment that
if $(x \mid y)=0$ for all $y \in H$, then $x=0$.
Indeed, the hypothesis implies that $(x \mid x)=0$, and then $x=0$ by the definition of an inner product.

When the inner product $(\cdot \mid \cdot)$ is understood we simply write $H$ instead of $(H,(\cdot \mid \cdot))$.

# 泛函分析代考

## 数学代写|泛函分析作业代写Functional Analysis代考|Hilbert Spaces

\begin{aligned} &\phi\left(v+v^{\prime}, w\right)=\phi(v, w)+\phi\left(v^{\prime}, w\right), \quad \phi(c v, w)=c \phi(v, w) \ &\phi\left(v, w+w^{\prime}\right)=\phi(v, w)+\phi\left(v, w^{\prime}\right), \quad \phi(v, c w)=\bar{c} \phi(v, w) \end{aligned}

(3.2) 中的筫共轭当然是多余的。

(i) $(x \mid x) \geqslant 0$ 对所有人 $x \in H$ 和 $(x \mid x)=0 \Rightarrow x=0$;
(二) $(x \mid y)=\overline{(y \mid x)}$ 对所有人 $x, y \in H$.

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