Definition:Pointwise Convergence

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Definition

Let $\sequence {f_n}$ be a sequence of real functions defined on $D \subseteq \R$.

Suppose that:

$\ds \forall x \in D: \lim_{n \mathop \to \infty} \map {f_n} x = \map f x$

That is:

$\forall x \in D: \forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n > N: \size {\map {f_n} x - \map f x} < \epsilon$


Then $\sequence {f_n}$ converges to $f$ pointwise on $D$ as $n \to \infty$.


(See the definition of convergence of a sequence).


Topological Space

Let $D$ be a set.

Let $T$ be a topological space.

For each $n \in \N$, let $f_n: D \to T$ be a mapping.

Let $f : D \to T$ be another mapping.


Then, $\sequence {f_n}$ converges pointwise to $f$ if and only if:

$\sequence {f_n}$ converges to $f$ in the product topology $T^D$


Also defined as

Some sources insist that $N \in \N$ but this is unnecessary and makes proofs more cumbersome.


Comment

Note that this definition of convergence of a function is weaker than that for uniform convergence, in which, given $\epsilon > 0$, it is necessary to specify a value of $N$ which holds for all points in the domain of the function.


In pointwise convergence, you need to specify a value of $N$ given $\epsilon$ for each individual point. That value of $N$ is allowed to be different for each $x \in D$.


Sources

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