Having moved through the preceding compactness ideas, we now study equicontinuous families in function spaces. The focus keyword for this lesson is equicontinuous families in function spaces. This page records the definitions, theorems, proofs, examples, and practice items needed for undergraduate work in compactness and its consequences in metric spaces. :::definition[Equicontinuous Family] A family $K\subseteq C[0,1]$ is $equicontinuous$ if for every $\varepsilon>0$ one $\delta>0$ works for all $f\in K$: $$\begin{equation*} |x-y|<\delta\implies |f(x)-f(y)|<\varepsilon. \end{equation*}$$ ::: :::example[Finite Families Are Equicontinuous] Every finite family in $C[0,1]$ is equicontinuous because finitely many uniform-continuity radii have a positive minimum. ::: :::theorem[Derivative Bound Implies Equicontinuity] If $|f'(t)|\leq M$ for all $f$ in a differentiable family and all $t\in[0,1]$, then the family is equicontinuous. ::: :::proof Given the derivative bound. To prove equicontinuity. The mean value theorem gives $|f(x)-f(y)|\leq M|x-y|$. Choose $\delta=\varepsilon/(M+1)$. $\square$ ::: :::example[Sine Family Is Not Equicontinuous] The family $\{\sin nx:n\in\mathbb{N}\}$ is uniformly bounded but not equicontinuous because $0$ and $\pi/(2n)$ approach each other while the function values differ by $1$. ::: :::exercise [1] State the main definition or theorem from this lesson. [2] Give one example where the hypothesis is used. [3] Explain one common mistake related to this topic. ::: :::answer [1] See the first formal block of the lesson. [2] The examples in the lesson provide standard choices. [3] A common mistake is to apply a Euclidean compactness rule in an arbitrary metric space. ::: :::faq[Frequently Asked Questions] Q: Why is the metric important? A: Compactness, boundedness, convergence, and openness are all interpreted using the chosen metric. Q: What should students check first? A: Identify the space, the metric, and the exact compactness criterion being used. ::: :::call-to-action[Continue to Uniform Boundedness in Function Spaces] subtitle: Review the definitions, proofs, and examples before moving to the next compactness idea. button: Next Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/uniform-boundedness-in-function-spaces button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/compact-neighbourhoods-and-exhaustions :::