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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.

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).

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.