Measure, Integration & Real Analysis, 2021a

210 Chapter 7 L p Spaces
18 Suppose (X, S, μ) is a measure space, 1 ≤ p, q ≤ ∞, and h : X → F is an
S-measurable function such that hf ∈ L q (μ) for every f ∈ L p (μ). Prove that
f ↦→ hf is a continuous linear map from L p (μ) to L q (μ).
A Banach space is called reflexive if the canonical isometry of the Banach space
into its double dual space is surjective (see Exercise 20 in Section 6D for the
definitions of the double dual space and the canonical isometry).
19 Prove that if 1 < p < ∞, then l p is reflexive.
20 Prove that l 1 is not reflexive.
21 Show that with the natural identifications, the canonical isometry of c 0 into its
double dual space is the inclusion map of c 0 into l ∞ (see Exercise 15 for the
definition of c 0 and an identification of its dual space).
22 Suppose 1 ≤ p < ∞ and V, W are Banach spaces. Show that V × W is a
Banach space if the norm on V × W is defined by
for f ∈ V and g ∈ W.
‖( f , g)‖ = ( ‖ f ‖ p + ‖g‖ p) 1/p
Measure, Integration & Real Analysis, by Sheldon Axler