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Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset ) necessarily contains at least one maximal element .
5 days ago · Zorn's Lemma. If is any nonempty partially ordered set in which every chain has an upper bound, then has a maximal element. This statement is equivalent to the axiom of choice . Renteln and Dundes (2005) give the following (bad) mathematical jokes about Zorn's lemma:
Zorn’s lemma, statement in the language of set theory, equivalent to the axiom of choice, that is often used to prove the existence of a mathematical object when it cannot be explicitly produced. In 1935 the German-born American mathematician Max Zorn proposed adding the maximum principle to the
Zorn’s lemma, because Zorn’s lemma in fact implies the axiom of choice. [Hint: Consider partially de ned choice functions suitably ordered, and use Zorn’s lemma to prove the existence of a maximal one. Then show that this maximal one is in fact globally de ned.]
We begin by proving a lemma that gives us the relation between real and imaginary parts of a complex linear functional. Lemma 3.1. Let Xbe a complex vector space and f: X! C a linear functional. There exists a real linear functional f 1: X! R such that f(x) = f 1(x) if(ix) for all x2X: Proof. Write f(x) = f
Zorn’s lemma is not intuitive, but it turns out to be logically equivalent to more readily appreciated statements from set theory like the Axiom of Choice (which says the Cartesian product of any family of nonempty sets is nonempty).
To apply Zorn’s lemma we only need to know that each chain has an upper bound in X; the upper bound need not be in the chain. The way we apply Zorn’s lemma in this note are
A partially ordered set in which every chain is bounded has a maximal element. Zorn's Lemma implies the Axiom of Choice. Given a surjection π: X → Y let P = {(B, β): B ⊂ Y, β: B → X with π ∘ β = IB}. Declare that (B1,β1) ≤ (B2,β2) if B1 ⊂ B2 and β2 ∣B1= β1.
Zorn’s lemma is a result in set theory that appears in proofs of some non-constructive existence theorems throughout mathematics. We will state Zorn’s lemma below and use it in later sections to prove results in group theory, ring theory, linear algebra, and topology. In an appendix we give an application of Zorn’s lemma to metric spaces.
Lecture 5: Zorn’s Lemma and the Hahn-Banach Theorem. Description: A first application of Zorn’s lemma is the existence of a Hamel basis. We then introduce the very useful Hahn-Banach theorem, which states that a bounded linear functional on a subspace can be continuously extended to the entire normed space.
