Abstract We establish inter alia a compactness criterion in metric spaces involving sequence of completely continuous mappings, which is continuously convergent, the sense H. Hahn, to identity mapping. For Banach spaces, linear version that result coincides with theorem due Mazur. also present new proof Ascoli–Arzelà theorem, we use above applied Bernstein operators.

It is well known that different kinds of expressions of modulus of convexity are essentially based on two geometrical propositions. For one of the propositions we first present a new proof by the Hahn–Banach theorem and intermediate value theorem, then give some corollaries to it which lead to some new expressions of modulus of convexity.

Abstract In this paper, we show new results and improvements of the non-Archimedean counterpart classical analysis in theory lineability. Besides analyzing algebraic genericity sets functions having properties regarding continuity, discontinuity, Lipschitzianity, differentiability analyticity, also study lineability sequences concerning boundedness convergence. particular (among several other r...

We characterize the class of separable Banach spaces X such that for every continuous function f : X → R and for every continuous function ε : X → (0,+∞) there exists a C smooth function g : X → R for which |f(x)− g(x)| ≤ ε(x) and g′(x) 6= 0 for all x ∈ X (that is, g has no critical points), as those infinite dimensional Banach spaces X with separable dual X∗. We also state sufficient condition...

In this paper, we give the concepts of properly distributed and simply distributed sequences, and prove that they are almost convergent. Basing on these, we review the work of Feng and Li [Feng, B. Q. and Li, J. L., Some estimations of Banach limits, J. Math. Anal. Appl. 323(2006) No. 1, 481-496. MR2262220 46B45 (46A45).], which is shown to be a special case of our generalized theory. 1. prelim...