A convex subset B of a real locally convex space B is said to have the separation property if it can be separated from every closed convex subset B of B , which is disjoint from B , by a closed hyperplane. The strong separation theorem says that if B is weakly compact, then it has the separation property. In this paper, we present two versions of the converse and discuss an application of them. For example, we prove that a normed space is reflexive if and only if its closed unit ball has the separation property. Results in this paper can be considered as supplements of the famous theorem of James.

Recently, Kawasaki and Takahashi (J. Nonlinear Convex Anal. 14:71-87, 2013) defined a broad class of nonlinear mappings, called widely more generalized hybrid, in a Hilbert space which contains generalized hybrid mappings (Kocourek <i>et al.</i> in Taiwan. J.Math. 14:2497-2511, 2010) and strict pseudo-contractive mappings (Browder and Petryshyn in J. Math. Anal. Appl. 20:197-228, 1967). They proved fixed point theorems for such mappings. In this paper, we prove fixed point theorems for widely more generalized hybrid non-self mappings in a Hilbert space by using the idea of Hojo <i>et al.</i> (Fixed Point Theory 12:113-126, 2011) and Kawasaki and Takahashi fixed point theorems (J.Nonlinear Convex Anal. 14:71-87, 2013). Using these fixed point theorems for non-self mappings, we proved the Browder and Petryshyn fixed point theorem (J.Math. Anal. Appl. 20:197-228, 1967) for strict pseudo-contractive

Using elementary arguments, we shall show that every bounded disjointness preserving linear functional of the Wiener ring L1 (Z) assumes the form (f)=

Let X be a compact Hausdorff space and C(X) the space of continuous functions defined on X. There are three versions of the Banach-Stone theorem. They assert that the Banach space geometry, the ring structure, and the lattice structure of C(X) determine the topological structure of X, respectively. In particular, the lattice version states that every disjointness preserving linear bijection T from C(X) onto C(Y) is a weighted composition operator Tf = h . f which provides a homeomorphism from Y onto X. In this note, we manage to use basically algebraic arguments to give this lattice version a short new proof. In this way, all three versions of the Banach-Stone theorem are unified in an algebraic framework such that different isomorphisms preserve different ideal structures of C(X).

Let A , A be two regular commutative unital Banach algebras such that A is integral over A . In 2003, Dawson and Feinstein showed that the topological stable rank A whenever A . In this note, we investigate whether we will have A in general. For instance, when A is a commutative unital A -algebra, we show that A , and the equality holds at least when the integral extension is separable. In general, A and A have the same Bass stable ranks A .