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In this paper, we give a number of characterizations for a Banach space X which is isometric to a subspace of c0, or, c0(Γ), successively, in terms of extreme points of its dual unit ball BX*, Frechet and Gâteaux derivatives of its norm, or, in terms of w*-strongly exposed points and w*-exposed points of BX*.

In this paper, we show that every super weakly compact convex subset of a Banach space is an absolute uniform retract, and that it also admits the uniform compact approximation property. These can be regarded as extensions of Lindenstrauss and Kalton’s corresponding results.

By characterizing Asplund operators through Frechet differentiability property of convex functions, we show the following Bishop-Phelps-Bollobes theorem: Suppose that X is a Banach space, T: X → C(K) is an Asplund operator with ∥T∥ = 1, and that x0 ∈ SX, 0 < e satisfy $$\left\| {T\left({{x_0}} \right)} \right\| > 1 - {{{\varepsilon^2}} \over 2}$$ . Then there exist xe ∈ SX and an Asplund operator S: X → C(K) of norm one so that $$\left\| {S\left({{x_\varepsilon}} \right)} \right\| = 1,\,\,\,\,\,\,\left\| {{x_0} - {x_\varepsilon}} \right\| < \varepsilon \,\,\,\,\,{\rm{and}}\,\,\,\,\,\,\,\left\| {T - S} \right\| < \varepsilon .$$ Making use of this theorem, we further show a dual version of Bishop-Phelps-Bollobas property for a strong Radon-Nikodym operator T: l1 → Y of norm one: Suppose that y0* ∈ Sy*, e ≥ 0 satisfy $$\left\| {{T^*}\left({y_0^*} \right)} \right\| > 1 - {{{\varepsilon^2}} \over 2}$$ . Then there exist y*µ ∈ Sy*, xe∈ (±en), ye∈Sy, and a strong Radon-Nikodym operator S: l1→ Y of norm one so that $$\begin{array}{*{20}{c}} {(i)}&{\left\| {S({x_\varepsilon })} \right\| = 1;}&{(ii)S({x_\varepsilon }) = {y_\varepsilon };}&{(iii)\left\| {T - S} \right\| < \varepsilon ;} \end{array}$$ $$\begin{array}{*{20}{c}} {(iv)\left\| {{S^*}(y_\varepsilon ^*)} \right\| = \left\langle {y_\varepsilon ^*,{y_\varepsilon }} \right\rangle = 1;}&{(v)\left\| {y_0^* - y_\varepsilon ^*} \right\| < \varepsilon }&{and}&{(vi)\left\| {{T^*} - {S^*}} \right\| < \varepsilon ,} \end{array}$$ where (en) denotes the standard unit vector basis of l1.

Suppose that X, Y are two real Banach Spaces. We know that for a standard ∈-isometry f: X → Y, the weak stability formula holds and by applying the formula we can induce a closed subspace N of Y*. In this paper, by using again the weak stability formula, we further show a sufficient and necessary condition for a standard ∈-isometry to be stable in assuming that N is w*-closed in Y*. Making use of this result, we improve several known results including Figiel’s theorem in reflexive spaces. We also prove that if, in addition, the space Y is quasi-reflexive and hereditarily indecomposable, then $$L(f) \equiv \overline {span} [f(x)]$$ contains a complemented linear isometric copy of X; Moreover, if X = Y, then for every ∈-isometry f : X → X, there exists a surjective linear isometry S : X → X such that f − S is uniformly bounded by 2∈ on X.

In this paper, we show that a closed convex subset C of a Banach space is strongly proximinal (proximinal, resp.) in every Banach space isometrically containing it if and only if C is locally (weakly, resp.) compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonempty set-valued and upper semi-continuous.

The purpose of this paper is to discuss those kinds of statistical convergence, in terms of filter F, or ideal I-convergence, which are equivalent to measure convergence defined by a single statistical measure. We prove a number of characterizations of a single statistical measure μ-convergence by using properties of its corresponding quotient Banach space l ∞/l ∞(I μ ). We also show that the usual sequential convergence is not equivalent to a single measure convergence.

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 < ɛ < 1/2. Then for every bounded linear operator T: H → H and x 0 ∈ H with ‖T‖ = 1 = ‖x 0‖ such that ‖Tx 0‖ > 1 − g3, there exist x ɛ ∈ H and a bounded linear operator S: H → H with ‖S‖ = 1 = ‖x ɛ ‖ such that $$\left\| {Sx_\varepsilon } \right\| = 1, \left\| {x_\varepsilon - x_0 } \right\| \leqslant \sqrt {2\varepsilon } + \sqrt[4]{{2\varepsilon }}, \left\| {S - T} \right\| \leqslant \sqrt {2\varepsilon } .$$

This paper presents two counterexamples about ball-coverings of Banach spaces and shows a new characterization of uniformly non-square Banach spaces via ball-coverings.

This paper presents a property of geometric and topological nature of Gateaux differentiability points and Frechet differentiability points of almost CL-spaces. More precisely, if we denote by M a maximal convex set of the unit sphere of a CL-space X, and by C M the cone generated by M, then all Gateaux differentiability points of X are just Un-s(C M ), and all Frechet differentiability points of X are ∪ int(C M ) (where n-s(C M ) denotes the non-support points set of CM).

To guarantee every real-valued convex function bounded above on a set is continuous, how ”thick” should the set be? For a symmetric set A in a Banach space E, the answer of this paper is: Every real-valued convex function bounded above on A is continuous on E if and only if the following two conditions hold: i) spanA has finite co-dimentions and ii) coA has nonempty relative interior. This paper also shows that a subset A ⊂ E satisfying every real-valued convex function bounded above on A is continuous on E if (and only if) every real-valued linear functional bounded above on A is continuous on E, which is also equivalent to that every real-valued convex function bounded on A is continuous on E.

This paper shows that every non-separable hereditarily indecomposable Banach space admits an equivalent strictly convex norm, but its bi–dual can never have such a one; consequently, every non–separable hereditarily indecomposable Banach space has no equivalent locally uniformly convex norm.

Every real-valued convex and locally Lipschitzian function f defined on a nonempty closed convex set D of a Banach space E is the local restriction of a convex Lipschitzian function defined on E. Moreover, if E is separable and int ⁡ D ≠ ∅ \operatorname {int} D \ne \emptyset , then, for each Gateaux differentiability point x ( ∈ int ⁡ D ) ( \in \operatorname {int} D) of f, there is a closed convex set C ⊂ int ⁡ D C \subset \operatorname {int} D with the nonsupport points set N ( C ) ≠ ∅ N(C) \ne \emptyset and with x ∈ N ( C ) x \in N(C) such that f C {f_C} (the restriction of f on C) is Fréchet differentiable at x.