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Revision difference : uniform convergence |
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Version 9 |
| Let $X$ be any set, and let $(Y,d)$ be a metric space. |
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 |
A sequence $f_1,f_2,\dots$ of functions mapping $X$ to $Y$ is said to be |
| \emph{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 $d(f_n(x),f(x))<\varepsilon$. |
\emph{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 $d(f_n(x),f(x))<\varepsilon$. |
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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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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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