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'uniform convergence'
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| Title of object: |
uniform convergence |
| Canonical Name: |
UniformConvergence |
| Type: |
Definition |
| Created on: |
2002-12-09 06:02:22.610554-05 |
| Modified on: |
2002-12-09 06:14:56.362042-05 |
| Defines: |
uniformly convergent, pointwise |
| Synonyms: |
uniform convergence=pointwise convergence |
Preamble:
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Content:
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 \textbf{pointwise convergent} (or simply convergent) to another function $f$, if the sequence $f_n(x)$ converges to $f(x)$ for each $x$; that is, if for each $x\in X$ and $\varepsilon>0$ there exists $N_x>0$ such that $d(f_n(x),f(x))<\varepsilon$ when $n>N_x$. This is usually denoted by $f_n\rightarrow f$.
The sequence of functions is said to be \textbf{uniformly convergent} if $N_x$ is independent of $x$; i.e. 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 $f_n\rightrightarrows f$. |
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