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This paper presents relations between several types of closedness of a law-invariant convex set in a rearrangement invariant space $\mathcal{X}$. In particular, we show that order closedness, $\sigma(\mathcal{X},\mathcal{X}_n^\sim)$-closedness and $\sigma(\mathcal{X},L^\infty)$-closedness of a law-invariant convex set in $\mathcal{X}$ are equivalent, where $\mathcal{X}_n^\sim$ is the order continuous dual of $\mathcal{X}$. We also provide some application to proper quasiconvex law-invariant functionals with the Fatou property., Comment: 10 pages

Let X be a Banach lattice. A well-known problem arising from the theory of risk measures asks when order closedness of a convex set in X implies closedness with respect to the topology $$\sigma (X,X_n^\sim )$$ , where $$X_n^\sim $$ is the order continuous dual of X. Motivated by the solution in the Orlicz space case, we introduce two relevant properties: the disjoint order continuity property (DOCP) and the order subsequence splitting property (OSSP). We show that when X is monotonically complete with OSSP and $$X_n^\sim $$ contains a strictly positive element, every order closed convex set in X is $$\sigma (X,X_n^\sim )$$ -closed if and only if X has DOCP and either X or $$X_n^\sim $$ is order continuous. This in turn occurs if and only if either X or the norm dual $$X^*$$ of X is order continuous. We also give a modular condition under which a Banach lattice has OSSP. In addition, we also give a characterization of X for which order closedness of a convex set in X is equivalent to closedness with respect to the topology $$\sigma (X,X_{uo}^\sim )$$ , where $$X_{uo}^\sim $$ is the unbounded order continuous dual of X.

We construct separable universal injective and projective lattices for the class of all separable Banach lattices.

An important consequence of the Hahn–Banach theorem says that on any locally convex Hausdorff topological space X, there are sufficiently many continuous linear functionals to separate points of X. In the paper, we establish a “local” version of this theorem. The result is applied to study the uo-dual of a Banach lattice that was recently introduced in Gao et al. (Positivity 22(3):711–725, 2018). We also provide a simplified approach to the measure-free characterization of uniform integrability established in Kardaras (J Funct Anal 266:1913–1927, 2014).

We provide a variety of results for (quasi)convex, law-invariant functionals defined on a general Orlicz space, which extend well-known results in the setting of bounded random variables. First, we show that Delbaen's representation of convex functionals with the Fatou property, which fails in a general Orlicz space, can be always achieved under the assumption of law-invariance. Second, we identify the range of Orlicz spaces where the characterization of the Fatou property in terms of norm lower semicontinuity by Jouini, Schachermayer and Touzi continues to hold. Third, we extend Kusuoka's representation to a general Orlicz space. Finally, we prove a version of the extension result by Filipovi\'{c} and Svindland by replacing norm lower semicontinuity with the (generally non-equivalent) Fatou property. Our results have natural applications to the theory of risk measures.

Let X and Y be metric spaces. A function \(f:X\rightarrow Y\) is said to be Lipschitz in the small if there are \(r> 0\) and \(K

Let X and Y be completely regular spaces and E and F be Hausdorff topological vector spaces. We call a linear map T from a subspace of C(X, E) into C(Y, F) a Banach–Stone map if it has the form T f (y) = S y (f (h(y))) for a family of linear operators S y : E → F, $${y \in Y}$$ , and a function h: Y → X. In this paper, we consider maps having the property: $$\bigcap^{k}_{i=1}Z(f_{i}) \neq\emptyset \iff \bigcap^{k}_{i=1}Z(Tf_{i})\neq\emptyset , \quad({\rm Z}) $$ where Z(f) = {f = 0}. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces of differentiable functions (including C ∞), as Banach–Stone maps. In particular, we confirm a conjecture of Ercan and Onal: Suppose that X and Y are realcompact spaces and E and F are Hausdorff topological vector lattices (respectively, C *-algebras). Let T: C(X, E) → C(Y, F) be a vector lattice isomorphism (respectively, *-algebra isomorphism) such that $$ Z(f) \neq\emptyset\iff Z(Tf) \neq\emptyset. $$ Then X is homeomorphic to Y and E is lattice isomorphic (respectively, C *-isomorphic) to F. Some results concerning the continuity of T are also obtained.

Suppose that (Fn)n=1∞ is a sequence of regular families of finite subsets of ℝ and (θn)n=1∞ is a nonincreasing null sequence in (0,1). The mixed Tsirelson spaceT[(θn,Fn)n=1∞] is the completion ofc00 with respect to the implicitly defined norm\(\left\| x \right\| = \max \{ \left\| x \right\|_{c_0 ,} \mathop {\sup }\limits_n \user2{sup}\theta _n \sum\limits_{i = 1}^k {\left\| {E_i x} \right\|} \} \), where the last supremum is taken over all sequences (Ei)i=1k in [ℕ]

The space WeakL 1 consists of all Lebesgue measurable functions on [0,1] such thatq(f)=supcλ{t:|f(t)|>c} c>0 is finite, where λ denotes Lebesgue measure. Let ρ be the gauge functional of the convex hull of the unit ball {f:q(f)≤1} of the quasi-normq, and letN be the null space of ρ. The normed envelope of WeakL 1, which we denote byW, is the space (WeakL 1/N, ρ). The Banach envelope of WeakL 1, $$\overline W $$ , is the completion ofW. We show that $$\overline W $$ is isometrically lattice isomorphic to a sublattice ofW. It is also shown that all rearrangement invariant Banach function spaces are isometrically lattice isomorphic to a sublattice ofW.

We give an example of a pair of Banach spacesE andF so that neitherE′ norF has cotype 2, but every bounded linear operator fromE intoF factors through Hilbert space.