Pointwise Convergence
A notion of convergence for sequences functions that reduces the question of convergence to convergence of sequences of real numbers.
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In the following, we consider sequences of functions and introduce some convergence concepts.
Definition 1 (Pointwise convergence). A sequence \((f_n)_{n\in\mathbb{N}}\) of functions \(f_n:I\to\mathbb{K}\) is called pointwisely convergent to \(f:I\to\mathbb{K}\) if for all \(x\in I\) holds \[\lim_{n\to\infty}f_n(x)=f(x).\] Using logical quantifiers this reads: \[\label{def:lptwconv} \forall x\in I \quad \forall \varepsilon > 0 \quad \exists N \in \mathbb{N} \quad \forall n \geq N \quad : \quad |f_n(x)-f(x)|<\varepsilon \ .\]
Pointwise convergence means nothing else but that for all \(x\in I\), the sequence \((f_n(x))_{n\in\mathbb{N}}\) in \(\mathbb{K}\) converges to \(f(x)\).
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