Constant convergence

Steady convergence ( English continuous convergence ) is a mathematical concept that both the functional analysis and as well as in numerical mathematics , and not least in approximation theory , the optimization theory and the calculus of variations is used. Associated with it are the concepts of uniform continuity and compact convergence .

definition

Given two arbitrary metric spaces and and a countable number of functions . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f,\;f_{1},f_{2},f_{3}\ldots \;\;\colon X\to Y}$

One then says that the sequence of functions is continuously convergent to ${\displaystyle (f_{n})_{n=1,2,\ldots }}$ ${\displaystyle f}$ if the following condition is met:

For and for each in convergent sequence in is always convergence . ${\displaystyle x\in X}$ ${\displaystyle X}$ ${\displaystyle x_{n}\rightarrow x\ (n\to \infty \;,\;x_{n}\in X)}$ ${\displaystyle Y}$ ${\displaystyle f_{n}(x_{n})\rightarrow f(x)\ (n\to \infty )}$

One then also says that the sequence of functions converges continuously to ${\displaystyle f_{n}}$ ${\displaystyle f}$ or something similar.

Connection of the terminology

The connection between continuous convergence, compact convergence and uniform continuity is shown by the following theorem :

For countably many functions of two metric spaces and hold and they are all continuous . ${\displaystyle f,f_{1},f_{2},f_{3}\ldots \;\colon X\to Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f_{n}{\xrightarrow {\text{punktweise}}}f\ (n\to \infty )}$ ${\displaystyle f_{n}}$

Then the following statements are equivalent:

(i) They form an equally continuous sequence of functions. ${\displaystyle f_{n}}$

(ii) is a continuous function and the sequence of functions converges continuously to . ${\displaystyle f}$ ${\displaystyle f}$

(iii) The sequence of functions converges compactly to . ${\displaystyle f}$

Dini's theorem

Last but not least, Dini's well-known sentence can be brought into the above context, which can be presented in an expanded version as follows:

Given a point-wise convergent and monotonic function sequence of real-valued continuous functions, whose limit function should also be continuous , on a metric space .

Then the convergence of this sequence of functions is continuous and uniform on every compact subset of metric space .

Continuous convergence on Banach spaces

As a further result from the above theorem one obtains a result for certain sequences of convex functionals on Banach spaces :

A Banach space and a convex area in it as well as a function sequence of continuous convex functions , which should converge point by point to a limit function , are given. ${\displaystyle X}$ ${\displaystyle U\subseteq X}$ ${\displaystyle f_{1},f_{2},f_{3}\ldots \;\colon U\to \mathbb {R} }$ ${\displaystyle f\colon U\to \mathbb {R} }$

Then:

(1) The limit function is convex and continuous.

(2) The sequence of functions converges continuously and compactly.

Remarks

• Continuous convergence always results in point-by-point convergence .
• Every Banach space is a normalized space and as such a complete space under the associated metric .

literature

• Peter Kosmol : Optimization and Approximation (=  De Gruyter Studies ). 2nd Edition. Walter de Gruyter & Co. , Berlin 2010, ISBN 978-3-11-021814-5 ( MR2599674 ).
• Peter Kosmol, Dieter Müller-Wichards : Optimization in Function Spaces . With stability considerations in Orlicz spaces (=  De Gruyter Series in Nonlinear Analysis and Applications . Volume 13 ). Walter de Gruyter & Co., Berlin 2011, ISBN 978-3-11-025020-6 ( MR2760903 ).

Individual evidence

1. ↑ Peter Kosmol: Optimization and Approximation. 2010, p. 69 ff.
2. ↑ Peter Kosmol, Dieter Müller-Wichards: Optimization in Function Spaces. 2011, p. 134 ff.
3. ↑ ^{a b c} Kosmol, op.cit., P. 71
4. ↑ ^{a b} Kosmol / Müller-Wichards, op.cit., P. 134
5. ↑ Kosmol, op. Cit., Pp. 336-337
6. ↑ Kosmol / Müller-Wichards, op.cit., P. 133
7. ↑ Kosmol, op.cit., P. 338