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I know both definitions but I was wondering what are the relations between them. My question is if someone could explain intuitively the differences between these types of convergence. Specifically, is it true that convergence in the uniform norm means uniform convergence.

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The way I like to think about it, uniform convergence means the whole function gets close, but in pointwise convergence, you can make any points in the two functions close, but some other part won't necessarily be close. The classic example is $x^n$ on [0,1] as n gets big. For most values of $x$, these functions approach 0, but the values near 1 will be a lot bigger. – Trurl Jun 8 '13 at 16:46
Thanks. Yes, but what about convergence in norm? Does convergence in norm by the uniform norm imply uniform convergence? – synack Jun 8 '13 at 16:59
up vote 1 down vote accepted
  1. Uniform convergence is the same as convergence in the uniform norm.
  2. Uniform convergence implies pointwise convergence.

Explanation of 1: being able to find $N$ such that $|f_n(x) - f(x)| \le \epsilon $ for all $x$ (in some interval) and for all $n\ge N$ is precisely the same as being able to find $N$ such that $\sup_x |f_n(x) - f(x)| \le \epsilon $ for all $n\ge N$.

Explanation of 2: pointwise convergence allows you to choose $N$ based on $x$, if you want. Uniform convergence requires same $N$ for all $x$. A more restrictive definition implies a less restrictive one.

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