什么是 complete metric space？

Example 4.8 (Cauchy without a limit)

In $X = \Q$ with the usual distance, the decimal truncations of $\sqrt2$,

satisfy $\abs{x_p - x_q} \leq 10^{-\min(p,q)}$: Cauchy in $\Q$. A limit in $\Q$ would also be the limit in $\R$, namely $\sqrt2 \notin \Q$: no limit exists in $X$. Incompleteness is the presence of such “phantom limits”; completeness of $\R$ was engineered in the Year 1 volume precisely to give every Cauchy sequence a home.

Example 4.13 (An integral equation)

On $X = C(\intcc{0}{1})$ (complete, Theorem 4.9), consider $T(f)(x) = 1 + \frac12 \int_0^x f(t)\,\dd t$. For $f, g \in X$:

so $T$ is a $\frac12$-contraction: it has a unique continuous fixed point — the solution of $f' = \frac f2$, $f(0) = 1$, namely $\eu^{x/2}$. This scheme, industrialized, becomes the Cauchy–Lipschitz theorem of Chapter 16.

Example 4.14 (A numerical fixed point: $x = \cos x$)

On the complete $X = \intcc{0}{1}$, the map $f = \cos$ sends $X$ into $\intcc{\cos 1}{1} \subseteq X$ and is a contraction: by the mean value inequality,

Banach: a unique solution of $x = \cos x$ in $\intcc01$ (hence in $\R$: any real fixed point lies in $\intcc{-1}{1}$, then in $\intcc{\cos 1}{1}$ after one application), and the iteration $x_{n+1} = \cos x_n$ converges to it from any start: $x_\infty \approx 0.739085$, the famous number obtained by hammering the cosine key of a calculator. The error bound predicts $(\sin1)^n/(1 - \sin1)$ decay — about one digit per $13$ presses; the a posteriori bound of this chapter’s weekend problem (question 14) certifies each step on the fly.

Example 7.5 (Perturbing the identity)

Let $g \colon \R^d \to \R^d$ be $k$-Lipschitz with $k < 1$. Then $\varphi = \mathrm{id} + g$ is a homeomorphism of $\R^d$ onto $\R^d$. Injectivity, with a quantitative modulus:

Surjectivity is the fixed point theorem: solving $\varphi(x) = y$ means $x = y - g(x)$, and $x \mapsto y - g(x)$ is a $k$-contraction of the complete $\R^d$ — a unique solution $x = \psi(y)$ exists for every $y$. The displayed inequality makes the inverse $\psi$ Lipschitz with constant $\frac1{1-k}$: a homeomorphism, with explicit bounds on both moduli. This innocuous-looking statement is the engine inside the inverse function theorem (Chapter 20): near a point where $Df$ is invertible, $f$ is an invertible linear map plus a small Lipschitz perturbation, and today’s example does the rest. It also quantifies numerical robustness: a system perturbed by less than the inverse’s margin remains solvable, with the solution moving by at most $\frac{1}{1-k}$ times the perturbation.