Measure, Integration & Real Analysis, 2021a

Section 6D Linear Functionals 179
Because our simplified form of Zorn’s Lemma deals with set inclusions rather
than more general orderings, we need to use the notion of the graph of a function.
6.67 Definition graph
Suppose T : V → W is a function from a set V to a set W. Then the graph of T
is denoted graph(T) and is the subset of V × W defined by
graph(T) ={ ( f , T( f ) ) ∈ V × W : f ∈ V}.
Formally, a function from a set V to a set W equals its graph as defined above.
However, because we usually think of a function more intuitively as a mapping, the
separate notion of the graph of a function remains useful.
The easy proof of the next result is left to the reader. The first bullet point
below uses the vector space structure of V × W, which is a vector space with natural
operations of addition and scalar multiplication, as given in Exercise 10 in Section 6B.
6.68 function properties in terms of graphs
Suppose V and W are normed vector spaces and T : V → W is a function.
(a) T is a linear map if and only if graph(T) is a subspace of V × W.
(b) Suppose U ⊂ V and S : U → W is a function. Then T is an extension of S
if and only if graph(S) ⊂ graph(T).
(c) If T : V → W is a linear map and c ∈ [0, ∞), then ‖T‖ ≤c if and only if
‖g‖ ≤c‖ f ‖ for all ( f , g) ∈ graph(T).
The proof of the Extension Lemma
(6.63) used inequalities that do not make
sense when F = C. Thus the proof of the
Hahn–Banach Theorem below requires
some extra steps when F = C.
Hans Hahn (1879–1934) was a
student and later a faculty member
at the University of Vienna, where
one of his PhD students was Kurt
Gödel (1906–1978).
6.69 Hahn–Banach Theorem
Suppose V is a normed vector space, U is a subspace of V, and ψ : U → F is a
bounded linear functional. Then ψ can be extended to a bounded linear functional
on V whose norm equals ‖ψ‖.
Proof First we consider the case where F = R. Let A be the collection of subsets
E of V × R that satisfy all the following conditions:
• E = graph(ϕ) for some linear functional ϕ on some subspace of V;
• graph(ψ) ⊂ E;
• |α| ≤‖ψ‖‖f ‖ for every ( f , α) ∈ E.
Measure, Integration & Real Analysis, by Sheldon Axler