## 数学代写|泛函分析作业代写Functional Analysis代考|Hausdorff’s Maximality Principle

The Hausdorff Maximality Principle is a statement that can be used to possibly extend arguments that work in the finite or countable case to sets of arbitrary size. There are a few proofs in this book that make use of this principle; it is only needed to extend results to “uncountably infinite” dimensions. As such, it is mainly of theoretical value, and this section can be skipped if the main interest is in applications.

Consider a collection $\mathcal{M}$ of subsets $M \subseteq X$ that satisfy a certain property $\mathcal{P}$. A chain $\mathcal{C}=\left{M_\alpha\right}$ of such sets is a nested sub-collection, meaning that for any two sets $M_\alpha, M_\beta \in \mathcal{C}$, either $M_\alpha \subseteq M_\beta$ or $M_\beta \subseteq M_\alpha$. A chain can contain any number of nested subsets, even uncountable. A chain is called maximal when it cannot be added to by the insertion of any subset in $\mathcal{M}$. Hausdorff’s maximality principle states that
Every chain in $\mathcal{M}$ is contained in some maximal chain in $\mathcal{M}$.
Hausdorff’s Maximality principle is often used to show there is a maximal set $E$ that satisfies some property $\mathcal{P}$ as follows: the empty chain can be extended to a maximal chain of sets $M_\alpha ;$ if it can be shown that the union of this chain $E:=\bigcup_\alpha M_\alpha$ also satisfies $\mathcal{P}$, then there are no sets properly containing $E$ which satisfy $\mathcal{P}$, by the maximality of $\left{M_\alpha\right}$, i.e., $E$ is a maximal set in $\mathcal{M}$.

At the end of this chapter, it is proved that Hausdorff’s maximality principle implies the Axiom of Choice. Conversely, the Hausdorff Maximality principle can be proved from the Axiom of Choice (using the other standard set axioms), i.e., they are logically equivalent to each other, as well as to a number of other formulations such as Zorn’s lemma and the Well-Ordering principle. These statements are not constructive in the sense that they give no explicit way of finding the choice function or the maximal chain, but simply assert their existence.

## 数学代写|泛函分析作业代写Functional Analysis代考|Metric and Vector Properties

By construction, normed spaces are metric spaces, as well as vector spaces. We can apply ideas related to both, in particular open/closed sets, convergence, completeness, continuity, connectedness, and compactness, as well as linear subspaces, linear independence and spanning sets, convexity, linear transformations, etc. Many of these notions have better characterizations in normed spaces, as the following propositions attest.
Proposition $7.8$
Proof Vector addition and the norm are in fact Lipschitz maps,
$\left|\left(x_1+y_1\right)-\left(x_2+y_2\right)\right| \leqslant\left|x_1-x_2\right|+\left|y_1-y_2\right|=\left|\left(x_1, y_1\right)-\left(x_2, y_2\right)\right|_{X^2}$,
$||x|-|y|| \leqslant|x-y|$.
Scalar multiplication is continuous: for any $\epsilon>0$, take $|\lambda-\mu|$ to be smaller than $\min (\epsilon / 3(1+|x|), 1)$ and $|x-y|<\min (\epsilon / 3(1+|\lambda|), 1)$, to get \begin{aligned}|\lambda x-\mu y| & \leqslant|\lambda x-\mu x|+|\mu x-\mu y| \ &=|\lambda-\mu||x|+|\mu||x-y| \ & \leqslant|\lambda-\mu||x|+|\lambda||x-y|+|\lambda-\mu||x-y| \ &<\epsilon . \end{aligned}

# 泛函分析代考

## 数学代写|泛函分析作业代写Functional Analysis代考|Hausdorff’s Maximality Principle

Hausdorff的极大原则指出，米包含在某个最大链中米.

## 数学代写|泛函分析作业代写Functional Analysis代考|Metric and Vector Properties

|(X1+是1)−(X2+是2)|⩽|X1−X2|+|是1−是2|=|(X1,是1)−(X2,是2)|X2,
||X|−|是||⩽|X−是|.

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