Pointwise Convergence

Last updated Nov 1, 2022

Let $X$ be a Set and let $(Y, \tau)$ be a Topological Space. Suppose $(f_{n})$ is a Sequence of Functions from $X$ to $Y$. Suppose $f: X \to Y$ so that $\lim\limits_{n \in \mathbb{N}} f_{n}(x) = f(x)$ $\forall x \in X$. Then we say that $(f_{n})$ converges pointwise to $f$.

1. We can also talk about Pointwise Convergence on subset $S \subset X$. This is just Pointwise Convergence of the Function Restriction on $S$.