Definition:Pointwise Limit

Definition

Let $S$ be a set.

Let $\sequence {f_n}_{n \mathop \in \N}$, $f_n: S \to \R$ be a sequence of real-valued functions.

Suppose that for all $s \in S$, the limit:

$\ds \lim_{n \mathop \to \infty} \map {f_n} s$

exists.

This definition needs to be completed. In particular: Incorporate infinite limits You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{DefinitionWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Then the pointwise limit of $\sequence {f_n}_{n \mathop \in \N}$, denoted $\ds \lim_{n \mathop \to \infty} f_n: S \to \R$, is defined as:

$\forall s \in S: \ds \map {\paren {\lim_{n \mathop \to \infty} f_n} } s := \lim_{n \mathop \to \infty} \map {f_n} s$

Pointwise limit thence is an instance of a pointwise operation on real-valued functions.

This definition needs to be completed. In particular: of course, makes sense in arbitrary metric/topological space. cover this You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{DefinitionWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

• Definition:Pointwise Operation on Real-Valued Functions