By a slight modification of the classical local Langlands correspondence, one can attach a locally algebraic representation of GL(n) to any n-dimensional potentially semistable Galois representation (with distinct Hodge-Tate weights). A conjecture of Breuil and Schneider asserts that the former admits an invariant norm. We will prove this when the latter comes from a classical point on an eigenvariety. More generally, for any definite unitary group G, we will explain how its eigenvariety (of some fixed tame level) mediates part of a global correspondence between Galois representations of the CM field, and Banach-Hecke modules B with a unitary G-action. For any regular weight W, we express the locally W-algebraic vectors of B in terms of the Breuil-Schneider representation on the Galois side.