In this paper we survey some recent results concerning separating polynomials on real Banach spaces. By this we mean a polynomial which separates the origin from the unit sphere of the space, thus providing an analog of the separating quadratic form on Hilbert space.
Let be the Banach space of all bounded and continuous functions on the closed unit ball of a complex Banach space X and holomorphic on the open unit ball, with sup norm, and let be the subspace of of those functions which are uniformly continuous on . A subset is a boundary for if for every . We prove that for X = d(w,1) (the Lorentz sequence space) and X = C₁(H), the trace class operators, there is a minimal closed boundary for . On the other hand, for X = , the Schreier space,...
We show that if U is a balanced open subset of a separable Banach space with the bounded approximation property, then the space ℋ(U) of all holomorphic functions on U, with the Nachbin compact-ported topology, is always bornological.
We characterize the holomorphic mappings between complex Banach spaces that may be written in the form , where is another holomorphic mapping and belongs to a closed surjective operator ideal.
This paper studies the relationship between the bidual of the (projective) tensor product of Banach spaces and the tensor product of their biduals.
When U is the open unit ball of a separable Banach space E, we show that , the predual of the space of bounded holomorphic mappings on U, has the bounded approximation property if and only if E has the bounded approximation property.
I will explain basic concepts/problems of complex analysis in infinite dimensions, and survey the few approaches that are available to solve those problems.