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The Hyers–Ulam–Aoki Type Stability of Some Functional Equations on Banach Lattices

Agbeko N. K.

Bulletin of the Polish Academy of Sciences. Mathematics

2015

Vol. 63, no. 2

177--184

In Agbeko (2012) the Hyers–Ulam–Aoki stability problem was posed in Banach lattice environments with the addition in the Cauchy functional equation replaced by supremum. In the present note we restate the problem so that it relates not only to supremum but also to infimum and their various combinations. We then propose some sufficient conditions which guarantee its solution.

An answer to a conjecture on AL-space

Ercan Z.

Demonstratio Mathematica

2006

Vol. 39, nr 3

597-598

We give an answer to a conjecture of Birnbaum in [3].

Reflexivity and the separable quotient problem for a class of Banach spaces

Wójtowicz M.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 4

383-394

Let E be a Banach lattice and let X be its closed subspace such that: X is complemented in E, or the norm of E is order continuous. Then X is reflexive iff X* contains no isomorphic copy of l1 iff for every n ≥ l, the nth dual X(n) of X contains no isomorphic copy of l1 iff X has no quotient isomorphic to c0 and X does not have a subspace isomorphic to l1 (Theorem 2). This is an extension of the results obtained earlier by Lozanovskiĭ, Tzafriri, Meyer-Nieberg, and Diaz and Fernández. The theorem is applied to show that many Banach spaces possess separable quotients isomorphic to one of the following spaces: c0, l1, or a reflexive space with a Schauder basis.