Basic Analysis – Gently Done Topological Vector Spaces

102 Basic Analysis
The next result we shall need is Hölder’s inequality.
Theorem 9.7 Let p > 1 and let q be such that 1 1
+ = 1 (q is called the
p q
exponent conjugate to p.) Then, for any x = (x n ) ∈ ℓ p and y = (y n ) ∈ ℓ q ,
∞�
n=1
|x n y n | ≤ �x� p �y� q .
If p = 1, the above inequality is valid if we set q = ∞.
Proof The case p = 1 and q = ∞ is straightforward, so suppose that p > 1.
Without loss of generality, we may suppose that �x�p = �y�q = 1. We then let
α = 1 1
, β =
p q , a = |xn |p , b = |yn | q and use the previous proposition.
Proposition 9.8 For any x = (x n ) ∈ ℓ p , with p > 1,
�x� p = sup{ � � ∞ �
n=1
x n y n
�
� : �y�q = 1}.
The equality also holds for the pairs p = 1 and q = ∞, and p = ∞, q = 1.
Proof Hölder’s inequality implies that the right hand side is not greater than the
lefthandside. Fortheconverse, consider y = (y n )withy n = sgnx n |x n | p/q /�x� p/q
if 1 < p < ∞, with y n = sgnx n if p = 1, and with y n = (δ nm ) m∈N , if p = ∞.
As an immediate corollary, we obtain Minkowski’s inequality.
Corollary 9.9 For any p ≥ 1 and x,y ∈ ℓ p , we have x+y ∈ ℓ p and
�x+y� p ≤ �x� p +�y� p .
Proof This follows from the triangle inequality and the preceding proposition.
Department of Mathematics King’s College, London