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Let X be a Banach space, u ∈ X** and K, Z two subsets of X**. Denote by d(u,Z) and d(K,Z) the distances to Z from the point u and from the subset K respectively. The Krein-Smulian Theorem asserts that the closed convex hull of a weakly compact subset of a Banach space is weakly compact; in other words, every w*-compact subset K ⊂ X** such that d(K,X) = 0 satisfies d(co^w*(K),X) = 0. We extend this result in the following way: if Z ⊂ X is a closed subspace of X and...

If X is a Banach space and C a convex subset of X*, we investigate whether the distance $d̂({\overline{co}}^{w*}\left(K\right),C):=supinf\left|\right|k-c\left|\right|:c\in C:k\in {\overline{co}}^{w*}\left(K\right)$ from ${\overline{co}}^{w*}\left(K\right)$ to C is M-controlled by the distance d̂(K,C) (that is, if $d̂({\overline{co}}^{w*}\left(K\right),C)\le Md̂(K,C)$ for some 1 ≤ M < ∞), when K is any weak*-compact subset of X*. We prove, for example, that: (i) C has 3-control if C contains no copy of the basis of ℓ₁(c); (ii) C has 1-control when C ⊂ Y ⊂ X* and Y is a subspace with weak*-angelic closed dual unit ball B(Y*); (iii) if C is a convex subset of X and X is considered canonically embedded into...

If X is a Banach space and C ⊂ X a convex subset, for x** ∈ X** and A ⊂ X** let d(x**,C) = inf||x**-x||: x ∈ C be the distance from x** to C and d̂(A,C) = supd(a,C): a ∈ A. Among other things, we prove that if X is an order-continuous Banach lattice and K is a w*-compact subset of X** we have: (i) $d̂({\overline{co}}^{w*}\left(K\right),X)\le 2d̂(K,X)$ and, if K ∩ X is w*-dense in K, then $d̂({\overline{co}}^{w*}\left(K\right),X)=d̂(K,X)$; (ii) if X fails to have a copy of ℓ₁(ℵ₁), then $d̂({\overline{co}}^{w*}\left(K\right),X)=d̂(K,X)$; (iii) if X has a 1-symmetric basis, then $d̂({\overline{co}}^{w*}\left(K\right),X)=d̂(K,X)$.

Let $X$ be a Banach space and $𝒮\mathrm{𝑒𝑞}\left(X^{**}\right)$ (resp., $X_{{\aleph }_0}$) the subset of elements $\psi \in X^{**}$ such that there exists a sequence ${\left(x_n\right)}_{n\ge 1}\subset X$ such that $x_n\to \psi$ in the $w^{*}$-topology of $X^{**}$ (resp., there exists a separable subspace $Y\subset X$ such that $\psi \in {\overline{Y}}^{w^{*}}$). Then: (i) if $Dens\left(X\right)\ge {\aleph }_1$, the property $X^{**}=X_{{\aleph }_0}$ (resp., $X^{**}=𝒮\mathrm{𝑒𝑞}\left(X^{**}\right)$) is ${\aleph }_1$-determined, i.e., $X$ has this property iff $Y$ has, for every subspace $Y\subset X$ with $Dens\left(Y\right)={\aleph }_1$; (ii) if $X^{**}=X_{{\aleph }_0}$, $(B\left(X^{**}\right),w^{*})$ has countable tightness; (iii) under the Martin’s axiom $\mathrm{𝑀𝐴}\left({\omega }_1\right)$ we have $X^{**}=𝒮\mathrm{𝑒𝑞}\left(X^{**}\right)$ iff $(B\left(X^{*}\right),w^{*})$ has countable tightness and $overline\text{co}\left(B\right)={\overline{\text{co}}}^{w^{*}}\left(K\right)$ for every subspace $Y\subset X$, every $w^{*}$-compact subset $K$ of $Y^{*}$, and every...