In the present paper, we study some properties of fuzzy norm of linear operators. At first the bounded inverse theorem on fuzzy normed linear spaces is investigated. Then, we prove Hahn Banach theorem, uniform boundedness theorem and closed graph theorem on fuzzy normed linear spaces. Finally the set of all compact operators on these spaces is studied.

We prove a ratio ergodic theorem for non-singular free Z and R actions, along balls in an arbitrary norm. Using a Chacon-Ornstein type lemma the proof is reduced to a statement about the amount of mass of a probability measure that can concentrate on (thickened) boundaries of balls in R. The proof relies on geometric properties of norms, including the Besicovitch covering lemma and the fact tha...

We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesgue-measure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages Anf do not converge to a computable element of L ([0, 1]). In particular, there is no computable bo...

We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesgue-measure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages Anf do not converge to a computable element of L ([0, 1]). In particular, there is no computable bo...

We prove strengthenings of the Birkhoff Ergodic Theorem for weakly mixing and strongly measure preserving systems. show that our pointwise theorem systems is strictly stronger than Wiener-Wintner Theorem. also Theorems characterize respectively. The methods this paper allow one to an enhanced ergodic other levels hierarchy such as ergodicity mild but not K-mixing. author plans include these add...