**Module leader**: Joel Moreira --- # 1. Introduction # 2. Normed spaces # 3 - 4. Metric spaces & continuity in metric spaces # 5 - 6. # 7. Compactness # 8. Connectedness # 9. Completeness Definition 9.1. [[Complete metric spaces]] See that completeness is not a topological property, i.e. complete metric space may be homeomorphic to incomplete metric space. Proposition 9.2. [[Subspace of a complete metric space is complete if and only if it is closed]]. ###### 9.2 Examples of complete metric spaces In [[MA141 Analysis 1]], we proved the [[Bolzano-Weierstrass theorem]], using the *monotone convergence theorem* to construct a convergent subsequence from any bounded sequence in $\mathbb{R}$. We then observed that every Cauchy sequence in $\mathbb{R}$ is bounded, so it must have a convergent subsequence. Finally, we showed that a Cauchy sequence must converge to the same limit as this subsequence, thereby proving that $\mathbb{R}$ is complete (see [[Completeness of real numbers]]). Proposition 9.3. [[Compact metric spaces are complete]], uses this same argument - **whenever a Cauchy sequence has a convergent subsequence, this forces the entire sequence to converge to the same limit** - without relying on the specific construction of the convergent subsequence. The converse is not true, e.g. $\mathbb{R}$ is complete but not compact. Theorem 9.4. $\mathbb{R}^{n}$ is complete wrt standard norm (see [[Product of complete spaces is complete]]) Theorem 9.5. $\ell^{p}$ is complete for $1\leq p \leq \infty$ Theorem 9.6. [[Completeness of the supremum norm on bounded real-valued functions on non-empty set]]. Theorem 9.7. [[Completeness of the supremum norm on bounded continuous real-valued functions on non-empty topological space]]. Corollary 9.8. [[Completeness of the supremum norm on continuous real-valued functions on non-empty compact set]]. ###### 9.3. Completion of metric spaces Definition 9.9. [[Completion of metric spaces]]. Theorem 9.10. Any metric space can be isometrically embedded into the Banach space $(B(X), \lVert \cdot \rVert_{\infty})$. Corollary 9.11. Completion of $X$ in $B(X)$ as closure of image of $X$ under isometry. ###### 9.4. The contraction mapping theorem Theorem 9.13. [[Contraction mapping theorem]] Theorem 9.14. [[Picard–Lindelöf theorem]]