# Isaacs I.M.'s Algebra. A graduate course PDF By Isaacs I.M.

ISBN-10: 0821847996

ISBN-13: 9780821847992

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Sample text

If α < f , then QF (α−1 f ) > 1. By the monotone convergence theorem, QF (α−1 fn ) → QF (α−1 f ), so Q(α−1 fn ) > 1 for n large and so, for n large, fn > α. It follows that lim inf fn ≥ f . (iii) Pick λ so that QF (λf ) < ∞. Then F (λ(f − fn )) ≤ F (λf ), so by the dominated convergence theorem, QF (λ(f − fn )) → 0. 9. 42 Convexity We are heading towards our next result that E (F ) is often separable. We begin with a calculation of χA for A a characteristic function. 11 Let A ⊂ M and χA its characteristic function.

Thus, f F = 1 and QF (f / f ) ≤ 12 < 1. We now turn to the main result on consequences of the Δ2 condition. 9 Let (M, μ) be a measure space with a nonatomic component and F a weak Young function. Then the following are equivalent: (i) F obeys the Δ2 condition. 7, is equivalent to) norm convergence. , L∞ is dense in L(F ) ). (iv) YF = L(F ) (v) YF is a vector space. 21) We will show (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (v) ⇒ (vi) ⇒ (i). 22) (we use μ(M ) = 1 here). 22), QF (2k g) ≤ 1 so g F ≤ 2−k . This shows if QF (gn ) → 0, then gn F → 0.

Note that since an open set is pseudo-open, ∪λ λU = X. If X is a Banach space with the norm topology picking U = {x | x ≤ 1}, B ⊂ λU if and only if supx∈B x ≤ |λ| so this extends the natural notion of bounded from the Banach space context. 2 Let X be a Banach space and let A be a set of elements of X ∗ , so for each x ∈ X, { (x) | ∈ A} is a bounded subset of K. 1) Remarks 1. This is just the Banach–Steinhaus principle. The direct proof is so short, we give it. The usual proof appeals to the Baire category theorem whose proof is essentially included in the proof below.