FIVE MAJOR RESULTS IN ANALYSIS AND TOPOLOGY Aaron ...

CHAPTER 4
The Hahn-Banach Theorem
Functional analysis encompasses, not surprisingly, the study of linear spaces and the set
of mappings whose domain is a linear space (including functionals). Three results are abso-
lutely necessary for the development of this field: the Open Mapping and Banach-Steinhaus
theorems which will be discussed in the next chapter, and the Hahn-Banach theorem, which
will be the focus of our current discussion. Before we can state the theorem, a number of
concepts and results need to be developed.
X,
Definition. Let X be a set. If ≤ is a binary relation on X such that, for every x,y,z ∈
(1) x ≤ x,
(2) x ≤ y and y ≤ x implies that x = y, and
(3) x ≤ y and y ≤ z implies that x ≤ z,
then ≤ is called a partial ordering of X. X, when given the relation ≤, is called a partially
ordered set. A chain in X is a sequence of elements x1,...,xn,... ∈ X such that x1 ≤ · · · ≤
xn ≤ · · ·.
Definition. If X is a partially ordered set, then x is called a maximal element in X if,
for each y ∈ X, x ≤ y if, and only if, x = y.
Axiom (Zorn’s Lemma). Let P be a partially ordered set in which every chain has an
upper bound. Then P possesses a maximal element.
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