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uniform convergence (Definition)

Let $ X$ be any set, and let $ (Y,d)$ be a metric space. A sequence $ f_1,f_2,\dots$ of functions mapping $ X$ to $ Y$ is said to be uniformly convergent to another function $ f$ if, for each $ \varepsilon>0$, there exists $ N$ such that, for all $ x$ and all $ n>N$, we have $ d(f_n(x),f(x))<\varepsilon$. This is denoted by $ f_n\xrightarrow[]{u} f$, or “ $ f_n\rightarrow f$ uniformly” or, less frequently, by $ f_n\rightrightarrows f$.



"uniform convergence" is owned by Koro.
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See Also: compact-open topology, converges uniformly

Also defines:  uniformly convergent

Attachments:
limit function of sequence (Theorem) by pahio
the limit of a uniformly convergent sequence of continuous functions is continuous (Theorem) by neapol1s
limit laws for uniform convergence (Theorem) by stevecheng
uniform convergence on union interval (Theorem) by pahio
point preventing uniform convergence (Theorem) by pahio
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Cross-references: mapping, functions, sequence, metric space
There are 27 references to this entry.

This is version 11 of uniform convergence, born on 2002-12-09, modified 2005-07-02.
Object id is 3700, canonical name is UniformConvergence.
Accessed 16481 times total.

Classification:
AMS MSC: 40A30 (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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