Convergenza uniforme

[domenico.migl]

Salve a tutti avrei bisogno di qualche chiarimento su questo teorema (Teorema sulla convergenza uniforme delle successioni):

Sia ${f_n}$ una successione di funzioni convergente puntualmente in $J sube I$ allora $f_n$ converge uniformemente ad $f$ se e solo se:
i. $exists nu in NN:Sup_(x in J) |f_n(x)-f(x)| in RR forall n > nu$
ii. $lim_(n to +infty) Sup_(x in J) |f_n(x)-f(x)|=0$

Non riesco bene a capire le due condizioni i ed ii... qualcuno potrebbe spiegarmele?

[javicemarpe]

By definition, a sequence $\{f_n\}$ converges to $f$ uniformly in $J$ if, given $\varepsilon>0$, we can find $n_0\in \mathbb{N}$ such that, for all $n\geq n_0$, we have $|f_n(x)-f(x)|<\varepsilon$ for all $x\in J$. This is equivalent to have an $n_0\in \mathbb{N}$ such that $\text{sup}_{x\in J}|f_n(x)-f(x)|<\varepsilon$ for all $n\geq n_0$, and this is de definition of $$\lim_{n\to\infty}\sup_{x\in J}|f_n(x)-f(x)|=0.$$

The conditions tell you that the difference between $f_n$ and $f$ in all the set $J$ is uniformly small (that is, all the differences in different points $x\in J$ are uniformly bounded by $\varepsilon$) for large values of $n$.

[domenico.migl]

Thanks! I understood it continuing to read my book.

[javicemarpe]

Oh, so I was late xD