Versions of Eberlein-Smulian and Amir-Lindenstrauss theorems in the framework of conditional sets

Abstract

Based on conditional set theory, we study conditional weak topologies, extending some well-known results to this framework and culminating with the proof of conditional versions of Eberlein-Šmulian and Amir-Lindenstrauss theorems. In pursuing this aim, we prove conditional versions of Baire Category theorem and Uniform Boundedness Principle.

Stability in locally L0-convex modules and a conditional version of James' compactness theorem

“…In [32,Definition 3.14] it was introduced the notion of conditional seminorm. In this setting where A is the measure algebra, we find that a seminorm · : E → R + defined on a conditional vector space E is a conditional function such that the generating stable function · :…”

“…For each x ∈ E, let us define the conditional function j x : E * → R with j x (x * ) = x * (x). Then, in [32,Theorem 3.5] was proved that j : E → E * * with j(x) := j x is a conditional isometry, i.e. x = j(x) for all x ∈ E, which is referred to as natural conditional embedding.…”

“…Then [U n ; n ∈ N] is a conditional neighborhood base of 0 ∈ E * * * for the conditional topology σ(E * * * , [x * * 0 , j(x n ) ; n ∈ N]). Now, the conditional version of Goldstine's theorem (see [32,Theorem 3.6]) claims that j * (B E * ) is conditionally σ(E * * * , E * * )-dense in B E * * * (where j * : E * → E * * * is the natural conditional embedding of E * ). In particular, for each n ∈ N there exists x * n ∈ B E * with j * (x * n ) ∈ U n .…”

“…This means that the conditional sequence {ρ * (y n )} is conditionally bounded, and we therefore have {y n } V ρ * (c), for some c ∈ R + . Then, the conditional weak compactness and the conditional Eberlein-Šmulian theorem (see [32,Theorem 4.7]) allows us to suppose that {y n } conditionally weakly converges to some y ∈ L 1 with y ≤ 0 and E [y | F] = −1. Consequently, we have liminf n ρ * (y n ) ≥ ρ * (y).…”

“…Let x * * ∈ E * * j(E) . The strategy of the proof of[32, Theorem 4.7] allows us to construct conditional sequences {x n } in K; {x * n } in E * ; and {n k } in N, which is conditionally increasing, so that y * * sup [|y * * (x * n )| ; n ∈ N], for all y * * ∈ [x * * , x * * − x n ; n ∈ N]|x * * (x * n ) − x * n (x k )| ≤ for all 1 ≤ n ≤ n k . (18)On the other hand, the conditional Banach-Alaoglu Theorem (see[8, Theorem 5.10]) yields that j(K) ω * is conditionally weakly- * compact.…”