21.2. HAHN BANACH THEOREM 543

Actually, |h(x)| ≤K ∥x∥ . The reason for this is that h(−x) =−h(x)≤K ∥−x∥=K ∥x∥ andtherefore, h(x)≥−K ∥x∥ so−h(x)≤K ∥x∥. Thus |h(x)| ≤K ∥x∥. Let F(x)≡ h(x)− ih(ix).By Lemma 21.2.5, F is complex linear.

Now wF(x) = |F(x)| for some |w|= 1. Therefore

|F(x)| = wF(x) = F (wx)≡ h(wx)−

must equal zero︷ ︸︸ ︷ih(iwx) = h(wx)

= |h(wx)| ≤ K∥wx∥= K ∥x∥ . ■

21.2.4 The Dual Space and Adjoint Operators

Definition 21.2.7 Let X be a Banach space. Denote by X ′ the space of continuouslinear functions which map X to the field of scalars. Thus X ′ = L (X ,F). By Theorem21.1.8 on Page 535, X ′ is a Banach space. Remember with the norm defined on L (X ,F),

∥ f∥= sup{| f (x)| : ∥x∥ ≤ 1}

X ′ is called the dual space.

Definition 21.2.8 Let X and Y be Banach spaces and suppose L ∈L (X ,Y ). Thendefine the adjoint map in L (Y ′,X ′), denoted by L∗, by

L∗y∗(x)≡ y∗(Lx)

for all y∗ ∈ Y ′.

The following diagram is a good one to help remember this definition.

X ′L∗

← Y ′

X→L

Y

This is a generalization of the adjoint of a linear transformation on an inner productspace from Linear Algebra. Recall

(Ax,y) = (x,A∗y)

What is being done here is to generalize this algebraic concept to arbitrary Banach spaces.There are some issues which need to be discussed relative to the above definition. First ofall, it must be shown that L∗y∗ ∈ X ′. Also, it will be useful to have the following lemmawhich is a useful application of the Hahn Banach theorem.

Lemma 21.2.9 Let X be a normed linear space and let x ∈ X \V where V is a closedsubspace of X. Then there exists x∗ ∈ X ′ such that x∗(x) = ∥x∥ ̸= 0, x∗ (V ) = {0}, and∥x∗∥= 1

dist(x,V ) ∥x∥ . In the case that V = {0} , ∥x∗∥= 1.

Proof: Let f :Fx+V→F be defined by f (αx+v)=α ∥x∥. First it is necessary to showf is well defined and continuous. If α1x+v1 = α2x+v2 then if α1 ̸= α2, then x ∈V whichis assumed not to happen so f is well defined. It remains to show f is continuous. Suppose