Let X be a Banach space with a basis. We prove the following characterizations:(i) X is finite-dimensional if and only if every power-bounded operator is uniformly ergodic.(ii) X is reflexive if and only if every power-bounded operator is mean ergodic.(iii) X is quasi-reflexive of order one if and only if for every power-bounded operator T, T or T g is mean ergodic.

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Abstract. This paper deals with generalizations of the notion of a polytope to infinite dimensions. The most general definition is the following: a bounded closed convex subset of a Banach space is called a polytope if each of its finite-dimensional affine sections is a (standard) polytope.We study the relationships between eight known definitions of infinite-dimensional polyhedrality. We provide a complete isometric classification of them, which gives solutions to several open problems. An almost complete isomorphic classification is given as well (only one implication remains open).