  Conjugate Gradient (CG) methods are one of the most effective iterative
methods to solve linear equations in Hilbert spaces. So far, they have been
inherently bound to these spaces since they make use of the inner product
structure. In more general Banach spaces one of the most prominent iterative
solvers are Landweber-type methods that essentially resemble the Steepest
Descent method applied to the normal equation. More advanced are subspace
methods that take up the idea of a Krylov-type search space, wherein an optimal
solution is sought. However, they do not share the conjugacy property with CG
methods. In this article we propose that the Sequential Subspace Optimization
(SESOP) method can be considered as an extension of CG methods to Banach
spaces. We employ metric projections to orthogonalize the current search
direction with respect to the search space from the last iteration. For the
l2-space our method then exactly coincides with the Polak-Ribi\`ere type of the
CG method when applied to the normal equation. We show that such an
orthogonalized search space still leads to weak convergence of the subspace
method. Moreover, numerical experiments on a random matrix toy problem and 2D
computerized tomography on lp-spaces show superior convergence properties over
all p compared to non-orthogonalized search spaces. This especially holds for
lp-spaces with small p. We see that the closer we are to an l2-space, the more
we recover of the conjugacy property that holds in these spaces, i. e., as
expected, the more the convergence behaves independently of the size of the
truncated search space.
