泛函分析全套课件（江泽坚）.pdfVIP

LI MIAO: LECTURES ON FUNCTIONAL ANALYSIS § Definition. Let be a set. Suppose there is a binary relation defined between certain pairs (a, b) of elements of , expressed by a ≺ b, with the properties: (O ) a ≺ a, ∀a ∈ (reflexive); 1 (O ) if a ≺ b and b ≺ a, then a = b (symmetric); 2 (O ) if a ≺ b and b ≺ c, then a ≺ c (transitive). 3 Then ≺ is called a partial ordering on and is called a par- tially ordered set. A partially ordered set is said to be totally ordered if for every pair (a, b) in , either a ≺ b or b ≺ a holds. Examples. 1. (, ≤) is a totally ordered set. 2. (P (A), ⊆) is a partially ordered set. 3. Let G be a group. Let S be the set of subgroups with the relation ′ ′ that H ≺ H if H is a subgroup of H . Then (S , ≺) is a partially ordered set. Given two subgroups, H, H′ of G, we do not necessarily have H ≺ H′ or H′ ≺ H . Let be a partially ordered set, and a subset. Then we can define a partial ordering on by defining that a ≺ b for a, b ∈ to hold if and only if a ≺ b in . We shall say that it is the partial ordering on induced by the ordering on . Definition. Let be a partially ordered set and be a subset of . An upper bound of (in ) is an element a ∈ such that b ≺ a for all b ∈ . By a maximal element m of one means an element such that if a ∈ and m ≺ a, then m = a. Zorn’s lemma. Let be a nonempty partially ordered set with the property that every totally ordered subset of has an upper bound in . Then contains at least one maximal element. It is known that Zorn’s le