638 CHAPTER 23. REPRESENTATION THEOREMS

Thus for t > 0,

Reφ (y) ≤ |1+ tφ (y)|−1t

∥x∥=1≤ ∥x+ ty∥−∥x∥

t+

o(t)t

and for t < 0,

Reφ (y)≥ |1+ tφ (y)|−1t

≥ ∥x+ ty∥−∥x∥t

+o(t)

tBy assumption that the directional derivative exists, and letting t→ 0+ and t→ 0−,

Reφ (y) = limt→0

∥x+ ty∥−∥x∥t

= ψ′y (0) .

Now φ (y) = Reφ(y)+ i Imφ(y) so φ(−iy) =−i(φ (y)) =−iReφ(y)+ Imφ(y) and

φ(−iy) = Reφ (−iy)+ i Imφ (−iy).

Hence Reφ(−iy) = Imφ(y). Consequently,

φ (y) = Reφ(y)+ i Imφ(y) = Reφ (y)+ iReφ (−iy)

= ψ′y(0)+ iψ ′−iy(0).

This proves the lemma when ∥φ∥= 1. For arbitrary φ ̸= 0, let φ (x) = ∥φ∥ ,∥x∥= 1. Thenfrom above, if φ 1 (y)≡ ∥φ∥

−1φ (y) , ∥φ 1∥= 1 and so from what was just shown,

φ 1 (y) =φ(y)∥φ∥

= ψ′y(0)+ iψ−iy(0) ■

This shows you can represent φ in terms of this directional derivative.Now here are some short observations. For t ∈ R, p > 1, and x,y ∈ C, x ̸= 0

limt→0

|x+ ty|p−|x|p

t= p |x|p−2 (RexRey+ Imx Imy)

= p |x|p−2 Re(x̄y) (23.9)

Also from convexity of f (r) = rp, for |t|< 1,

|x+ ty|p−|x|p ≤ ∥x|+ |t| |y∥p−|x|p

=

[(1+ |t|)

(|x|+ |t| |y|

1+ |t|

)]p

−|x|p ≤ (1+ |t|)p |x|p

1+ |t|+|t| |y|p

1+ |t|− |x|p

≤ (1+ |t|)p−1 (|x|p + |t| |y|p)−|x|p ≤((1+ |t|)p−1−1

)|x|p +2p−1 |t| |y|p

Now for f (t) ≡ (1+ t)p−1 , f ′ (t) is uniformly bounded, depending on p, for t ∈ [0,1] .Hence the above is dominated by an expression of the form

Cp (|x|p + |y|p) |t| (23.10)

The above lemma and uniform convexity of Lp can be used to prove a general versionof the Riesz representation theorem next. Let p > 1 and let η : Lq→ (Lp)′ be defined by

η(g)( f ) =∫

Ω

g f dµ. (23.11)