In this article, we give a definition of a functional space which is constructed from all continuous functions defined on a compact topological space. We prove that this functional space is a Banach algebra. Next, we give a definition of a function space which is constructed from all real-valued continuous functions with bounded support. We prove that this function space is a real normed space.

There are traditionally many interactions between the convex geometry community and the Banach space community. In recent years, work is being done as well on problems that are related to notions and concepts from other fields. The interaction of convex geometry and Banach space theory, and also with other areas, is due to high dimensional phenomena which lie at the crossroad of convex geometry...

We give some sufficient conditions for proper lower semicontinuous functions on metric spaces to have error bounds (with exponents). For a proper convex function f on a normed space X the existence of a local error bound implies that of a global error bound. If in addition X is a Banach space, then error bounds can be characterized by the subdifferential of f . In a reflexive Banach space X, we...

Let X be a Banach space and let S(X)= {x ∈ X , ‖x‖ = 1} be the unit sphere of X . Parameters E(X) = sup{α(x), x ∈ S(X)}, e(X) = inf{α(x), x ∈ S(X)}, F(X) = sup{β(x), x ∈ S(X)}, and f (X) = inf{β(x), x ∈ S(X)}, where α(x) = sup{‖x + y‖2 + ‖x − y‖2, y ∈ S(X)}, and β(x) = inf{‖x + y‖2 + ‖x− y‖2, y ∈ S(X)} are introduced and studied. The values of these parameters in the lp spaces and function spac...

We prove that the dual or any quotient of a separable reflexive Banach space with the unconditional tree property has the unconditional tree property. This is used to prove that a separable reflexive Banach space with the unconditional tree property embeds into a reflexive Banach space with an unconditional basis. This solves several long standing open problems. In particular, it yields that a ...

We show that when a linear quotient map to a separable Banach space X has a Lipschitz right inverse, then it has a linear right inverse. If a separable space X embeds isometrically into a Banach space Y , then Y contains an isometric linear copy of X. This is false for every nonseparable weakly compactly generated Banach space X. Canonical examples of nonseparable Banach spaces which are Lipsch...

We present a connected version of the compact L-space constructed by Kenneth Kunen under CH. show that this provides Corson space K such Banach C(K) is isomorphic to no continuous function on zero-dimensional compactum.