Metric Spaces and Completeness

Metric Spaces: Completeness, Contraction, and Functional Spaces

Metric: Minimal Structure for Analysis

A metric space $(X, d)$ is a set $X$ with a distance function $d: X \times X \rightarrow \mathbb{R}_{\ge 0}$ satisfying: $d(x,y) = 0 \iff x = y$; $d(x,y) = d(y,x)$; $d(x,z) \le d(x,y) + d(y,z)$ (the triangle inequality).

Examples of metrics:

Euclidean: $d(x,y) = |x - y|$ on $\mathbb{R}^n$.

Taxicab (Manhattan): $d(x, y) = \sum |x_i - y_i|$ on $\mathbb{R}^n$.

Discrete: $d(x,y) = 1$ if $x \neq y$, $0$ if $x = y$.

Functional: $d(f,g) = \max |f(x) - g(x)|$ on $C[a,b]$ — uniform convergence.

$p$-adic: $|n|_p = p^{-v_p(n)}$, where $v_p$ is the exponent of $p$ in the factorization of $n$. Used in number theory and cryptography.

Completeness

$(X, d)$ is complete if every Cauchy sequence ($d(x_m, x_n) \to 0$ as $m,n \to \infty$) converges to an element of $X$.

Examples: $\mathbb{R}$ is complete (completeness axiom). $\mathbb{Q}$ is not complete: the sequence $3, 3.1, 3.14, 3.141, \ldots$ is Cauchy in $\mathbb{Q}$ but converges to $\pi \notin \mathbb{Q}$. $C[a,b]$ with the uniform norm is complete (Banach space).

Completion: Every metric space $X$ can be densely embedded into a unique complete space $\overline{X}$ (completion). The completion of $\mathbb{Q}$ is $\mathbb{R}$. The completion of $C([0,1])$ with respect to $L^2$-norm is $L^2([0,1])$.

Baire's Principle: A complete metric space is of the second category (not a countable union of nowhere dense sets). Corollary: $\mathbb{R}$ is uncountable (if $\mathbb{R} = \bigcup_n {x_n}$, then each ${x_n}$ is nowhere dense $\rightarrow$ contradiction).

Principle of Contracting Mappings (Banach, 1922)

Theorem: If $T: X \rightarrow X$ on a complete $(X, d)$, and $d(Tx, Ty) \leq q d(x, y)$ for $q < 1$ (contracting mapping), then $T$ has a unique fixed point $x^$: $Tx^ = x^$. Iterations: $x_{n+1} = T x_n \rightarrow x^$ exponentially ($d(x_n, x^) \leq q^n d(x_0, x^)$).

Proof: ${x_n}$ is a Cauchy sequence (estimate through $q^n \rightarrow 0$). Limit $x^* = \lim x_n$. Continuity of $T$: $Tx^* = T(\lim x_n) = \lim T x_n = \lim x_{n+1} = x^*$. Uniqueness: if $Tx = x$ and $Ty = y$, then $d(x, y) = d(Tx, Ty) \leq q d(x, y) \rightarrow d(x, y) = 0$.

Numerical Applications of the Contraction Principle

Picard Theorem (ODE): $y'(t) = f(t, y(t)), y(t_0) = y_0$. Operator $T\varphi = y_0 + \int_{t_0}^t f(s, \varphi(s)) ds$. If $f$ is Lipschitz in the second argument: $T$ is a contraction on $C([t_0, t_0 + \delta])$ with the uniform norm. Fixed point $=$ solution of the ODE!

Numerical example: $y' = y, y(0) = 1$. $T\varphi = 1 + \int_0^t \varphi(s) ds$. Iterations: $\varphi_0 = 1$, $\varphi_1 = 1 + t$, $\varphi_2 = 1 + t + t^2/2$, $\varphi_3 = 1 + t + t^2/2 + t^3/6$. Limit: $e^t$ — the Taylor series converges to the unique solution.

Newton's Method: For $f(x) = 0$: $T x_n = x_n - f(x_n) / f'(x_n)$. Locally $T$ is a contraction ($q \sim 0$) $\rightarrow$ quadratic convergence: the number of correct digits doubles at each step.

Spaces $C[a,b]$ and $L^p$

$C[a,b]$ with $|f|_\infty = \max |f|$: Banach (complete normed) space. Completeness $=$ uniform convergence of continuous functions preserves continuity.

$C[a,b]$ with $L^2$-norm $|f|_2 = \sqrt{\int f^2 dx}$: Not complete! Completion $=$ $L^2[a,b]$ (square integrable functions with respect to Lebesgue measure). $L^2$ is a Hilbert space (inner product $\langle f, g \rangle = \int f g dx$).

$L^p[a,b] = {f: \int |f|^p dx < \infty}$, norm $|f|_p = (\int |f|^p dx)^{1/p}$: Complete (by the Riesz–Fischer theorem). Duality: $(L^p)^* \cong L^q$, $1/p + 1/q = 1$. Hölder's inequality: $|\int f g| \leq |f|_p |g|_q$.

Sobolev Spaces $H^k = {f \in L^2: D^\alpha f \in L^2 \text{ for } |\alpha| \leq k}$: Foundation of modern theory of PDEs. Solutions of PDEs are sought in $H^k$ — "weak solutions".

Real Applications

Numerical analysis: Simple iteration method $x_{n+1} = g(x_n)$ converges $\iff$ $g$ contracts. Convergence criterion: $|g'(x^)| < 1$. Newton's method: $g = x - f/f'$, $g' = f f'' / (f')^2 \rightarrow$ at $f(x^) = 0$: $g'(x^*) = 0 \rightarrow$ "superlinear" (quadratic) convergence.

Machine Learning: Gradient descent — iterations: $w_{n+1} = w_n - \eta \nabla L(w_n)$. With suitable $\eta$: contracts $\rightarrow$ converges. RNN (recurrent neural networks) — iterations of the state transition operator; exploding gradient $=$ operator does not contract.

Complete Metric Spaces in Differential Equations Theory

Completeness of metric spaces is a fundamental property substantiating existence and uniqueness of solutions in mathematical physics problems. Banach's theorem on contracting mappings (the contraction principle) underlies Picard's method of successive approximations for the Cauchy problem for ODEs: with the Lipschitz condition on the right-hand side, Picard's iterations converge in the complete space $C([0,T])$ to a unique solution. This same completeness of $C([0,T])$ guarantees the validity of numerical methods like Runge–Kutta and Adams: they approximate Picard's iterations and converge precisely by virtue of the space's completeness. Sobolev spaces $W^{k,p}(\Omega)$ — completions of $C^\infty(\Omega)$ with a norm that takes derivatives up to order $k$ into account — are complete by construction. "Weak solutions" of elliptic and parabolic partial differential equations reside in them: heat equation, wave equation, Navier–Stokes equation. The Lax–Milgram lemma (existence and uniqueness of weak solution) works exactly in Hilbert (complete) spaces. In digital signal processing, Schauder bases in $L^2([0,1])$ — orthonormal Daubechies wavelets — are used for image compression in the JPEG 2000 standard.

Task: (a) Prove that $T(x) = \cos(x)$ on $[0,1]$ is a contraction. Find the fixed point (solution to $x = \cos x$) by iteration: start with $x_0 = 0.5$, perform 4 iterations. (b) Is $C[0,1]$ with $|f|1 = \int_0^1 |f| dt$ a complete space? Provide an example of a Cauchy sequence without a limit in $C[0,1]$. (c) Space $\ell^\infty = {(x_n){n\ge 1}: \sup |x_n| < \infty}$ with the sup norm — prove that it is complete.