# Is uniform convergence and pointwise convergence the same in function space C[X,R]?

In GF Simmon's book on Topology, I came across the definitions of point wise convergence of a sequence of functions and uniform convergence of a sequence of functions.

For all future references, $$C[X,R]$$ is the set of continuous bounded real functions defined on the metric space $$X,$$ and $$B$$ is the subset of all bounded real functions on $$X.$$

Let $$\{f_n\}$$ be a sequence of real functions defined on $$X.$$

Pointwise convergence is when if for each point $$x\in X,$$ the sequence of real functions $$\{f_n(x)\}$$ converges to a function $$f(x)$$, i.e., at each $$x,$$ for a given $$\varepsilon$$, there exists a natural $$N$$ such that for all $$n>N,$$ $$|f_n(x)-f(x)|<\varepsilon$$.

Uniform Convergence is when the $$\varepsilon$$ doesn't depend on $$x.$$ Which means that a natural $$N$$ can be found for each $$\varepsilon$$ such that it works for all $$x\in X.$$

In terms of convergence of the sequence of functions to a function, Pointwise means that for a given $$\varepsilon$$ you will find an $$N$$ for each $$x\in X.$$ Uniform means that for a given $$\varepsilon$$ you will find an $$N$$ which works for all $$x\in X.$$ Am I right?

• Writing $f_n(x)$-$f(x)$ rather than $f_n(x)-f(x)$ is not proper MathJax usage. See my edits to this question. – Michael Hardy Nov 21 at 18:14
• Dear @MichaelHardy, Thanks for your kindness. I am learning. – Krishan Nov 22 at 6:14

Well, strictly speaking, the quantifiers in your last paragraph are not okay. Actually, it should be: "Pointwise mean that for a $$\varepsilon>0$$, and for each $$x\in X$$, there exists an $$N\in\mathbb N$$ such that...". Can you see the difference? We talk first about the point $$x\in X$$. Your understanding of uniform convergence is correct.
With respect to the title of you question, these kind of convergence are not the same, even when $$X$$ is compact. Consider $$X=[0,1]$$, the sequence $$f_n(x)=x^n$$ converge pointwise to a function which is not even continuous, despite of that, the sequence does not have a uniform limit (since the uniform limit of continuous functions is continuous).
• There is no difference between "$\forall \varepsilon > 0$, $\forall x \in X$, ..." and "$\forall x \in X$, $\forall \varepsilon > 0$..." The quantified definition of pointwise convergence in the OP's post is correct. – TheSilverDoe Nov 21 at 18:44
• He wrote: "you will find an $N$ for each $x\in X$". That can be written as $\exists N\forall x$, which is not the same that $\forall x\exists N$. – Emmanuel C. Nov 21 at 18:57
• "You will find an $N$ for each $x \in X$". is not $\exists N, \forall x$. That would be "you will find an $N$ such that for each $x \in X$"... – TheSilverDoe Nov 21 at 19:37