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The first part of this paper is devoted to the theory of Poisson-Lie groupsin the Banach setting. Our starting point is the straightforward adaptation ofthe notion of Manin triples to the Banach context. The difference with thefinite-dimensional case lies in the fact that a duality pairing between twonon-reflexive Banach spaces is necessary weak (as opposed to a strong pairingwhere one Banach space can be identified with the dual space of the other). Thenotion of generalized Banach Poisson manifolds introduced in this paper iscompatible with weak duality pairings between the tangent space and a subspaceof the dual. We investigate related notion like Banach Lie bialgebras andBanach Poisson-Lie groups, suitably generalized to the non-reflexive Banachcontext.  The second part of the paper is devoted to the treatment of particularexamples of Banach Poisson-Lie groups related to the restricted Grassmannianand the KdV hierarchy. More precisely, we construct a Banach Poisson-Lie groupstructure on the unitary restricted Banach Lie group which acts transitively onthe restricted Grassmannian. A "dual" Banach Lie group consisting of (a classof) upper triangular bounded operators admits also a Banach Poisson-Lie groupstructure of the same kind. We show that the restricted Grassmannian inherits ageneralized Banach Poisson structure from the unitary Banach Lie group, calledBruhat-Poisson structure. Moreover the action of the triangular BanachPoisson-Lie group on it is a Poisson map. This action generates the KdVhierarchy, and its orbits are the Schubert cells of the restrictedGrassmannian.

Banach Poisson-Lie groups and Bruhat-Poisson structure of the restricted Grassmannian