Having moved through the preceding compactness ideas, we now study compactness criteria in sequence spaces. The focus keyword for this lesson is compactness criteria in sequence 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[Sequence Space Metric] For $X=\ell^p$, $p\geq1$, define $$\begin{equation*} d(x,y)=\left(\sum_{i=1}^{\infty}|x_i-y_i|^p\right)^{1/p}. \end{equation*}$$ ::: :::example[Sequence Space Is Not Compact] The unit vectors $e_n$ in $\ell^p$ satisfy $d(e_m,e_n)=2^{1/p}$ for $m\neq n$, so they have no Cauchy subsequence. Hence $\ell^p$ is not compact. ::: :::theorem[Total Boundedness Criterion in Sequence Spaces] A subset of $\ell^p$ is totally bounded if and only if it is bounded and its tails are uniformly small beyond a common index. ::: :::proof Given that $Y\subseteq\ell^p$. To prove the criterion. Finite nets force all tails to be uniformly controlled by the tails of finitely many net points. Conversely, finite-dimensional truncations are totally bounded, and the uniform tail estimate controls the remaining coordinates. $\square$ ::: :::solved-problem The Hilbert cube $\{x\in\ell^2:|x_n|\leq1/n\}$ is compact because it is closed, complete, bounded, and has uniformly small tails. ::: :::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 Compactness in Discrete and Euclidean 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/compactness-in-discrete-and-euclidean-spaces button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/arzela-ascoli-compactness-criterion :::