Abstract. Let K be a Hausdorff space and C b (K) be the Banach algebra of all complex bounded continuous functions on K. We study the Gâteaux and Fréchet differentiability of subspaces of C b (K). Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace H of C b (K) is a dense G δ subset of H, if K is compact. This gives a generalized Bishop's theorem, which says that the closure of the set of strong peak point for H is the smallest closed norming subset of H. The classical Bishop's theorem was proved for a separating subalgebra H and a metrizable compact space K.In the case that X is a complex Banach space with the Radon-Nikodým property, we show that the set of all strong peak functions inX is holomorphic} is dense. As an application, we show that the smallest closed norming subset of A b (B X ) is the closure of the set of all strong peak points for A b (B X ). This implies that the norm of A b (B X ) is Gâteaux differentiable on a dense subset of A b (B X ), even though the norm is nowhere Fréchet differentiable when X is nontrivial. We also study the denseness of norm attaining holomorphic functions and polynomials. Finally we investigate the existence of numerical Shilov boundary. IntroductionLet K be a Hausdorff topological space. A function algebra A on K will be understood to be a closed subalgebra of C b (K) which is the Banach algebra of all bounded complex-valued continuous functions on K. The norm f of a bounded continuous function f on K is defined to be sup x∈K |f (x)|. A function algebra A is called separating if for two distinct points s, t in K, there is f ∈ A such that f (s) = f (t).In this paper, a subspace means a closed linear subspace. For each t ∈ K, let δ t be an evaluation functional onis called separating if for distinct points t, s in K we have αδ t = βδ s for any complex numbers α, β of modulus 1 as a linear functional on A. This definition of a separating subspace is a natural extension of the definition of a separating function algebra. In fact, given a separating function algebra A 2000 Mathematics Subject Classification. 46B04, 46G20, 46G25, 46B22.

We introduce property (quasi-α), which implies property (A) defined by Lindenstrauss [10] and whose dual property is property (quasi-β) [2]. We consider relations between this property and other sufficient conditions for property (A), and study the denseness of norm attaining mappings under the conditions of these properties. In particular, if each of the Banach spaces X k , 1 ≤ k ≤ n − 1, has property (quasi-α) and Xn has property (A), then the projective tensor product X1 ⊗ π · · · ⊗ π Xn has property (A).

Abstract. We show that there is a unique norm-preserving extension for normattaining 2-homogeneous polynomials on the predual d * (w,

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Let X and Y be Banach spaces. Let P ( n X : Y ) be the space of all Y -valued continuous n-homogeneous polynomials on X. We show that the set of all norm-attaining elements is dense in P ( n X : Y ) when a set of u.s.e. points of the unit ball BX is dense in the unit sphere SX . Applying strong peak points instead of u.s.e. points, we generalize this result to a closed subspace of