12.3. HAHN BANACH THEOREM 311

Lemma 12.3.4 Suppose V is a complex normed linear space and f : V → R satisfiesa f (x) = f (ax) for all a real and f (x+ y) = f (x)+ f (y) . Define F (x) ≡ f (x)− i f (ix).Then F is linear on V with field of scalars equal to C.

Proof: Obviously F (x+ y) = F (x)+F (y). Now

F ((a+ ib)x) ≡ f ((a+ ib)x)− i f (i(a+ ib)x) = f (ax+ ibx)− i f (−bx+ iax)

= a f (x)+b f (ix)+bi f (x)−ai f (iax)

(a+ ib)F (x)≡ (a+ ib)( f (x)− i f (ix) = a f (x)− ia f (ix))+ ib f (x)+b f (ix)

Thus F is linear as claimed. ■

Corollary 12.3.5 (Hahn Banach) Let M be a subspace of a complex normed linearspace X, and suppose f : M→ C is linear and satisfies | f (x)| ≤ K ∥x∥ for all x ∈M. Thenthere exists a linear function F, defined on all of X such that F(x) = f (x) for all x ∈M and|F(x)| ≤ K ∥x∥ for all x.

Proof: Since | f (x)| ≤ K ∥x∥ for all x ∈M, then |Re f (x)| ≤ K ∥x∥ on M and so, sinceRe f is real and real linear on M, Re f (x)≤ K ∥x∥ ≡ ρ (x). By the Hahn Banach theorem,let h be a real valued linear extension of Re f satisfying h(x) ≤ K ∥x∥ on X . Now letF (x) ≡ h(x)− ih(ix) so F is complex linear on X . For a given x ∈ X , there is α ∈ C,|α|= 1 such that αF (x) = |F (x)| . Then

|F (x)|= αF (x) = F (αx) = h(αx)−=0︷ ︸︸ ︷

ih(iαx) = h(αx)≤ K ∥αx∥= K ∥x∥ ■

12.3.3 The Dual Space and Adjoint Operators

Definition 12.3.6 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 Theorem2.9.4 on Page 69, 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 12.3.7 Let X and Y be Banach spaces and L ∈L (X ,Y ). Then definethe 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, the conjugate transpose. Recall (Ax,y) = (x,A∗y) . What is being done here is togeneralize this algebraic concept to arbitrary Banach spaces. There are some issues whichneed to be discussed relative to the above definition. First of all, it must be shown thatL∗y∗ ∈ X ′. Also, it will be useful to have the following lemma which is a useful applicationof the Hahn Banach theorem.