Locally uniform convergence

The locally uniform convergence is a mathematical term that describes a certain type of convergence of function sequences and weakens the concept of uniform convergence . This term, which is closely related to compact convergence, plays an important role in analysis , as it acquires properties such as continuity or holomorphism .

Concept formation

Let it be a sequence of functions in a topological space as well as another function . One says that the sequence converges locally uniformly to if there is an open neighborhood of at every point , so that , that is, if the constraints of on there converge uniformly to the constraint of on . ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f_{n}\colon X\rightarrow \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle f\colon X\rightarrow \mathbb {R} }$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f}$ ${\displaystyle x\in X}$ ${\displaystyle U_{x}}$ ${\displaystyle x}$ ${\displaystyle \sup _{y\in U_{x}}|f_{n}(y)-f(y)|\rightarrow 0}$ ${\displaystyle f_{n}}$ ${\displaystyle U_{x}}$ ${\displaystyle f}$ ${\displaystyle U_{x}}$

Generalizations

A first obvious generalization is obtained by replacing the target space with a standardized space and the amount on with the associated norm . This applies in particular to the normalized space with the absolute value as the norm. This gives the concept of locally uniform convergence of complex-valued functions, which is important for function theory. ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

The next step is to replace the norm with a set of semi-norms and demand for each of these semi-norms , whereby the environment may also depend on. With this one can consider the locally uniform convergence of function sequences with values ​​in a locally convex space . After all, you don't need semi-norms in the target area; half-metrics are sufficient , that is, the expression is replaced by , a system of half-metrics running through. Any uniform room can therefore be used as a general target area . ${\displaystyle \sup _{y\in U_{x}}p(f_{n}(y)-f(y))\rightarrow 0}$ ${\displaystyle p}$ ${\displaystyle U_{x}}$ ${\displaystyle p}$ ${\displaystyle f_{n}\colon X\rightarrow Y}$ ${\displaystyle p(f_{n}(y)-f(y))}$ ${\displaystyle d(f_{n}(y),f(y))}$ ${\displaystyle d}$

Finally, you can replace the sequence with a network and get: ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (f_{i})_{i\in I}}$

Let it be a topological space, a uniform space whose uniformity is given by a system of half-metrics, a network of functions and a function. converges evenly to locally if there is an open neighborhood for every half-metric and every point such that . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle (f_{i})_{i\in I}}$ ${\displaystyle f_{i}\colon X\rightarrow Y}$ ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle (f_{i})_{i\in I}}$ ${\displaystyle f}$ ${\displaystyle d\in {\mathcal {D}}}$ ${\displaystyle x\in X}$ ${\displaystyle U_{x}\subset X}$ ${\displaystyle \sup _{y\in U_{x}}d(f_{i}(y),f(y))\rightarrow 0}$

Important applications

Follow continuous functions

• Limit values ​​of locally uniformly convergent sequences of continuous functions are again continuous.

This theorem is more general than the corresponding theorem on uniform convergence, for example converges locally uniformly but not uniformly. ${\displaystyle e^{x}=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}}=\lim _{n\to \infty }\sum _{k=0}^{n}{\frac {x^{k}}{k!}}}$

Follow holomorphic functions

• Weierstraß's theorem of convergence : Limits of locally uniformly convergent sequences of holomorphic functions are again holomorphic.

This theorem is important in function theory. Note that a corresponding proposition in real theory, that is, for functions that can be differentiated as often as desired, is wrong.

Comparison with the compact convergence

The compact convergence follows from the locally uniform convergence . If namely a sequence of functions which locally converges uniformly to one another and is compact , then there is an open neighborhood for each , so that there is uniform convergence on this neighborhood. Since is compact, one can already cover a finite number of these , and it follows and with it the claimed compact convergence. (The proof for networks of functions with values ​​in uniform spaces can be done in the same way.) ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f_{n}\colon X\rightarrow \mathbb {R} }$ ${\displaystyle f}$ ${\displaystyle K\subset X}$ ${\displaystyle x\in X}$ ${\displaystyle U_{x}\subset X}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle U_{x}}$ ${\displaystyle \sup _{y\in K}|f_{n}(y)-f(y)|\rightarrow 0}$

The converse generally does not apply, but it does in locally compact spaces , because in these, by definition, every point has an environment whose closure is compact.

Since large parts of analysis and function theory take place in locally compact spaces and local uniform convergence and compact convergence coincide there, a clear distinction is not always made between the two concepts of convergence. It should therefore be noted that the above-cited theorem about locally uniform limit values ​​of continuous functions for compact convergence is generally wrong, as an example on the Arens-Fort space shows (see there).

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• Hans Grauert, Wolfgang Fischer: Differential and integral calculus II , Springer Berlin Heidelberg New York (1978), ISBN 3-540-08697-8
• Wolfgang Fischer, Ingo Lieb: Function theory , Friedr Vieweg & Sohn Verlagsgesellschaft mbH (1980), ISBN 3-528-07247-4