Zorn's lemma

Definition: (Partial Order)
A partial order over set is a binary relation such that

Definition: Comparable
are said to be comparable if either or .

Definition: Totally ordered
A partially ordered set is said to be totally ordered if every two elements are comparable.

Definition: Greatest element
Suppose is a partially ordered set. is called the greatest element of if

Definition: Upper bound of a subset.
, is called an upper bound of , if

Definition: Maximal element
is called the maximal element if

极大元一定是最大元, 反之不对

Theorem: Zorn's lemma
Suppose is a partially ordered set in which every totally ordered subset has an upper bound. Then has a maximal element.

Existence of Hamel Basis of a vector space

Suppose , define

We can define a partial order on ,

Let be a totally ordered subset in . Then

is a linear independent subset. In fact, for all , there is such that

Since , is comparable and thus there exists largest element, we assume it is without loss of generality. Thus . Since , is linear independent and thus are linear independent. Now we have , and

This implies any totally ordered subset has an upper bound. Now the zorn's lemma implies there exists an maximal point .

Now we will show that is a basis. Suppose otherwise, there exists such that can not be written by a finite linear combination of elements of , then we consider

Easy to verify that and . Since is an maximal element, we have , contradicts.