Single-uniform convergence

The simple-uniform convergence is a concept of convergence from the mathematical branch of analysis . It is a weakening of the uniform convergence . The term was defined by Ulisse Dini, among others .

definition

Be a subset. A sequence of functions that converges pointwise is said to converge to simply-uniformly if ${\displaystyle D_{f}\subset \mathbb {R} }$ ${\displaystyle (f_{n}\colon D_{f}\to \mathbb {R} )_{n\in \mathbb {N} }}$ ${\displaystyle f}$

${\displaystyle \forall \varepsilon >0\ \exists N\in \mathbb {N} \ \forall x\in D_{f}:\operatorname {Card} (\{n\mid n\geq N,\ \left|f_{n}(x)-f(x)\right|<\varepsilon \})=\aleph _{0}}$

applies. With is meant the power of . ${\displaystyle \aleph _{0}}$ ${\displaystyle \mathbb {N} }$

properties

Every uniformly convergent sequence of functions is also simply-uniformly convergent.

Individual evidence

1. ^ EW Hobson: The Theory of Functions of a Real Variable and the Theory of Fourier's Series. 2nd edition, Cambridge, ISBN 978-1418186517 , pp. 105-106.