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In my lecture notes it gives the following definition for pointwise and uniform convergence:

The sequence $f_k$ converges pointwise to a function $f$ on $A$ as $k \rightarrow\infty$ if the sequence $(f_k(x))_k\epsilon_\mathbb{N})$ converges to $f(x)$ for every $x\in A$; namely if for every $x \in A$ and every $\epsilon >0$ there exists $N\in \mathbb{N}$ such that for every $k \geq N$ we have $|f_k(x)-f(x)|<\epsilon$.

The sequence $f_k$ converges uniformly to $f$ on $A$ if $\forall \epsilon>0$ $\exists N \in \mathbb{N}$ such that $\forall k \geq N, \forall x\in A: |f_k(x)-f(x)|<\epsilon$

I'm confused about the difference between the definition of pointwise and uniform convergence. They both seem the same to me. Could anyone clarify the difference please?

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    $\begingroup$ The difference is that the N in uniform applies to every single x in A. For pointwise, the choice of N may depend on x. $\endgroup$
    – DaveNine
    Nov 11, 2016 at 17:49

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The key here is your $k.$ In pointwise convergence your $k \equiv k(x,\epsilon)$ but for uniform convergence it depends only on $\epsilon.$

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Intuitively, uniform convergence is like having a tubular neighbourhood of the limit function and the fuctions $f_n$ are definitely in the tube.

With pointwise convergence we simply ask that the succession $a_n:=f_n(x)$, $x$ fixed, tends to $f(x)$.

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