By A.M. Fink

ISBN-10: 3540067299

ISBN-13: 9783540067290

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**Extra resources for Almost Periodic Differential Equations**

**Sample text**

N→∞ Then f is a norm one linear functional on c. It is clear that f ≤ p on c. 4 (the Hahn–Banach Extension Theorem), there exists a linear functional f˜ : ∞ → R such that f˜|c = f and f˜ ≤ p. 10, it can be shown that f˜ ∈ 1 . It is worth noting that f˜ has an additional property: If ξ is a bounded sequence of nonnegative real numbers, then f˜(ξ ) ≥ 0. To see this, suppose that ξ = (ξn )∞ n=1 is a bounded sequence such that ξn ≥ 0 for all n ∈ N. Then p(−ξ ) = sup (−ξn ) ≤ 0. n∈N By construction, f˜ ≤ p on ∞ , and so f˜(−ξ ) ≤ 0.

Suppose now that p(−x) = −p(x) for all x ∈ E. Then, by the triangle inequality, p(x + y) = −p(−x − y) ≥ − (p(−x) + p(−y)) = p(x) + p(y). ✷ The reverse inequality is simply subadditivity of p. Let PE be the collection of all sublinear functionals on a real vector space E. We define an order on the set PE by saying p ≤ q whenever p(x) ≤ q(x) for all x ∈ E. 6 Suppose p ∈ PE and V is a linear subspace of a real vector space E. If q is a sublinear functional on V such that q ≤ p|V , then there exists a sublinear functional r ∈ PE such that r|V = q and r ≤ p.

Xn ) Show that the norms · (b) Show that p ⊆ q , but p q p = |x1 |p + · · · + |xn |p 1/p . and · q are equivalent on Rn . is not a subset of p . 11 Let x ∈ r for some r < ∞. Show that x ∈ that x p → x ∞ as p → ∞. 12 Suppose (Ω, μ) is a positive measure space and let 1 ≤ p < q ≤ ∞. (a) Prove that if μ(Ω) < ∞, then f p ≤ Cp,q f q for all measurable functions f , where Cp,q is a constant that depends on p and q. (b) Show that the assumption μ(Ω) < ∞ cannot be omitted in (a). (c) Find a real-valued function f on [0, 1] such that f p < ∞ but f q = ∞.

### Almost Periodic Differential Equations by A.M. Fink

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