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converges uniformly (Definition)

Let $X$ be a set, $(Y,\rho)$ a metric space and $\{f_n\}$ a sequence of functions from $X$ to $Y$, and $f\colon X\to Y$ another function.

If for every $\varepsilon>0$ there exists an integer $N$ such that

\begin{displaymath}\rho(f_n(x),f(x))<\varepsilon \end{displaymath}

for all $x\in X$ and all $n>N$, then we say that $f_n$ converges uniformly to $f$.



"converges uniformly" is owned by yark. [ full author list (4) | owner history (3) ]
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See Also: uniform convergence, convergent series

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Cross-references: integer, functions, sequence, metric space
There are 36 references to this entry.

This is version 7 of converges uniformly, born on 2003-10-15, modified 2006-09-17.
Object id is 4986, canonical name is ConvergesUniformly.
Accessed 7419 times total.

Classification:
AMS MSC40A30 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences of functions)

Pending Errata and Addenda
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duplication by polarbear on 2007-09-11 11:35:23
this is the same with "uniform convergence" entry, only the title is different(which could have been added as "also defines").
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classification by yark on 2004-01-19 05:17:49
40A30
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