A conjugate direction method for linear systems in Banach spaces

In this article, the well-known conjugate gradient (CG) method for linear systems in Hilbert spaces is extended to a reflexive Banach space setting. In this setting, the Riesz isomorphism has to be replaced by the duality mapping. Due to the nonlinearity of the duality mapping, the short term recursion and conjugacy of search directions cannot be maintained simultaneously. The well-posedness of the proposed iteration and its global convergence are shown under appropriate conditions. Error bounds and stopping criteria are presented as well. The results extend to a limited-memory variant of the algorithm. The behavior of the method is demonstrated by numerical examples.

An elementary approach to a lattice-valued Banach-Stone Theorem

Let $ X $ and $ Y $ be compact Hausdorff spaces, and $ E $ be a nonzero real Banach lattice. In this note, we give an elementary proof of a lattice-valued Banach-Stone theorem by Cao, Reilly and Xiong [3] which asserts that if there exists a Riesz isomorphism $ \Phi: C(X,E)\rightarrow C(Y,\mathbb{R}) $ such that $ \Phi(f) $ has no zeros if $ f $ has none, then $ X $ is homeomorphic to $ Y $ and $ E $ is Riesz isomorphic to $ \mathbb{R} $.