Having moved through the preceding compactness ideas, we now study uniform boundedness in function spaces. The focus keyword for this lesson is uniform boundedness 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[Uniformly Bounded Family] A family $K\subseteq C[0,1]$ is $uniformly bounded$ if there exists $M>0$ such that $|f(x)|\leq M$ for all $x\in[0,1]$ and all $f\in K$. ::: :::example[Finite Families Are Uniformly Bounded] A finite family is uniformly bounded by taking the maximum of the individual bounds. ::: :::theorem[Pointwise Bounded Equicontinuous Sequences Are Uniformly Bounded] A pointwise bounded equicontinuous sequence of real-valued functions on $[0,1]$ is uniformly bounded. ::: :::proof Given pointwise boundedness and equicontinuity. To prove uniform boundedness. Define $g(x)=\sup_n |f_n(x)|$. Equicontinuity makes $g$ continuous. Since $[0,1]$ is compact, $g$ is bounded, giving one common bound. $\square$ ::: :::example[Uniformly Bounded but Not Equicontinuous] The functions $f_n(x)=x^2/(x^2+(1-nx)^2)$ are bounded by $1$, but $f_n(0)=0$ and $f_n(1/n)=1$. ::: :::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 Arzela-Ascoli Compactness Criterion] 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/arzela-ascoli-compactness-criterion button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/equicontinuous-families-in-function-spaces :::