# convergence in function space

Maybe is a silly question, but for some reason I am confused...

If $\mathcal{F}$ is a normed space of real functions and $\displaystyle{ f \in \mathcal{\bar F } }$ then there exists a sequence of functions $\displaystyle{ (f_n ) \subset \mathcal{F} }$ such that $\displaystyle{ f_n \to f \quad \text{as} \quad n \to \infty}$ which is equivelant to $\displaystyle{ || f_n -f|| \to 0 \quad \text{as} \quad n \to \infty}$

Here is my question: The convergence $f_n \to f$ is uniform or pointwise ?

Thank's in advance!

edit: $\displaystyle{ f, f_n : A \subset \mathbb R \to \mathbb R }$

Is now more clear my question?

Any ideas?

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1
What norm do you have? – enzotib Oct 4 '12 at 17:56
@enzotib: Some norm in general. It depends on the norm...??? – passenger Oct 4 '12 at 17:58

## 1 Answer

Convergence $f_n\to f$ here is defined by $\|f_f-f\|\to 0$ (whereas the latter is just convergence in $\mathbb{R}$). This notion can indeed lead to very different notions of convergence of functions.

A simple example is convergence with respect to the supremum norm $\|f\|_\infty=\sup|f(x)|$ and the 1-norm $\|f\|_1 = \int|f(x)|dx$. The former is just uniform convergence, while the latter is a notion of convergence which is usually not treated in calculus courses. In fact the latter allows unbounded sequences of functions which still converge to the zero function (and you should try to find such an example by yourself).

If I remember correctly, "pointwise almost everywhere convergence" is not induced by a norm...

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Thank you for your reply! I missed to write that $f, f_n : \mathbb R \to \mathbb R$. So you say that the type of convergence ( uniform or pointwise) depends on the norm we use? – passenger Oct 4 '12 at 18:25
The important thing here is: There are more notions of convergence of sequences of functions than just uniform and pointwise. Uniform convergence is induced by the sup-norm while pointwise convergence is not characterized by any norm (if I remember correctly). But there are a lot more notions for convergence of sequences of functions. – Dirk Oct 5 '12 at 6:18
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