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Normal structure and weakly normal structure of Orlicz spaces

Shutao Chen, Yanzheng Duan (1991)

Commentationes Mathematicae Universitatis Carolinae

Every Orlicz space equipped with Orlicz norm has weak sum property, therefore, it has weakly normal structure and fixed point property. A criterion of sum property also of normal structure for such spaces is given as well, which shows that every Orlicz space has isonormal structure.

Normal structure of Lorentz-Orlicz spaces

Pei-Kee Lin, Huiying Sun (1997)

Annales Polonici Mathematici

Let ϕ: ℝ → ℝ₊ ∪ 0 be an even convex continuous function with ϕ(0) = 0 and ϕ(u) > 0 for all u > 0 and let w be a weight function. u₀ and v₀ are defined by u₀ = supu: ϕ is linear on (0,u), v₀=supv: w is constant on (0,v) (where sup∅ = 0). We prove the following theorem. Theorem. Suppose that Λ ϕ , w ( 0 , ∞ ) (respectively, Λ ϕ , w ( 0 , 1 ) ) is an order continuous Lorentz-Orlicz space. (1) Λ ϕ , w has normal structure if and only if u₀ = 0 (respectively, ∫ v ₀ ϕ ( u ₀ ) · w < 2 a n d u ₀ < ∞ ) . (2) Λ ϕ , w has weakly normal structure if and only if ∫ 0 v ₀ ϕ ( u ₀ ) · w < 2 .

Norm-attaining polynomials and differentiability

Juan Ferrera (2002)

Studia Mathematica

We give a polynomial version of Shmul'yan's Test, characterizing the polynomials that strongly attain their norm as those at which the norm is Fréchet differentiable. We also characterize the Gateaux differentiability of the norm. Finally we study those properties for some classical Banach spaces.

Normed algebras of differentiable functions on compact plane sets: completeness and semisimple completions

Heiko Hoffmann (2011)

Studia Mathematica

We continue the study of the completeness and completions of normed algebras of differentiable functions Dⁿ(K) (where K is a perfect, compact plane set), initiated by Bland, Dales and Feinstein [Studia Math. 170 (2005) and Indian J. Pure Appl. Math. 41 (2010)]. We prove new characterizations of the completeness of D¹(K) and results concerning the semisimplicity of the completion of D¹(K). In particular, we prove that semi-rectifiability is necessary for the completion of D¹(K) to be semisimple in...

Normed barrelled spaces

Christopher Stuart (2000)

Banach Center Publications

In this paper we present a general “gliding hump” condition that implies the barrelledness of a normed vector space. Several examples of subspaces of l 1 are shown to be barrelled using the theorem. The barrelledness of the space of Pettis integrable functions is also implied by the theorem (this was first shown in [3]).

Normed "upper interval" algebras without nontrivial closed subalgebras

C. J. Read (2005)

Studia Mathematica

It is a long standing open problem whether there is any infinite-dimensional commutative Banach algebra without nontrivial closed ideals. This is in some sense the Banach algebraists' counterpart to the invariant subspace problem for Banach spaces. We do not here solve this famous problem, but solve a related problem, that of finding (necessarily commutative) infinite-dimensional normed algebras which do not even have nontrivial closed subalgebras. Our examples are incomplete normed algebras rather...

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