Complement

Fix a universe set UUU (an ambient set containing all sets under discussion). The complement of A⊆UA\subseteq UA⊆U (relative to UUU) is Ac:=U∖A={x∈U:x∉A}.A^c := U\setminus A = \{x\in U : x\notin A\}.Ac:=U∖A={x∈U:x∈/A}.Complements are fundamental in topology and measure theory because “closed” is defined as “complement of open,” and many set identities are most naturally expressed using complements. ...

Complete metric space

A metric space (X,d)(X,d)(X,d) is complete if every Cauchy sequence in XXX converges to a limit in XXX. Formally: ∀(xn)⊆X, [(xn) Cauchy ⇒ ∃x∈X with xn→x]. \forall (x_n)\subseteq X,\ \Bigl[(x_n)\ \text{Cauchy}\ \Rightarrow\ \exists x\in X\ \text{with}\ x_n\to x\Bigr]. ∀(xn)⊆X, [(xn) Cauchy ⇒ ∃x∈X with xn→x].Completeness is a core structural property in analysis: it is needed for many fixed-point and category arguments and is the metric analogue of completeness of R\mathbb{R}R as an ordered field. ...

Completeness axiom of R

The completeness axiom (least upper bound property) for R\mathbb{R}R states: If E⊆RE\subseteq \mathbb{R}E⊆R is nonempty and bounded above , then EEE has a least upper bound in R\mathbb{R}R; that is, there exists s∈Rs\in\mathbb{R}s∈R such that x≤sx\le sx≤s for all x∈Ex\in Ex∈E (so sss is an upper bound), and if uuu is any upper bound of EEE, then s≤us\le us≤u (so sss is the least upper bound), and we write s=sup⁡Es=\sup Es=supE. Completeness is the crucial axiom separating R\mathbb{R}R from Q\mathbb{Q}Q and is the source of many convergence and compactness results in real analysis. ...

Completeness equivalences

The real numbers R\mathbb{R}R are complete . In practice, completeness can be expressed in several equivalent ways. Completeness equivalences (standard list): The following statements are equivalent (each can be taken as a definition of completeness of R\mathbb{R}R): Least upper bound property: Every nonempty set E⊆RE\subseteq\mathbb{R}E⊆R that is bounded above has a supremum in R\mathbb{R}R. Cauchy completeness: Every Cauchy sequence in R\mathbb{R}R converges to a real number. Nested interval property: If In=[an,bn]I_n=[a_n,b_n]In=[an,bn] are nested closed intervals with In+1⊆InI_{n+1}\subseteq I_nIn+1⊆In and bn−an→0b_n-a_n\to 0bn−an→0, then ⋂n=1∞In\bigcap_{n=1}^\infty I_n⋂n=1∞In consists of exactly one point. Monotone convergence: Every bounded monotone sequence in R\mathbb{R}R converges. These equivalences explain why different-looking arguments (suprema, Cauchy sequences, nested intervals, monotone sequences) are interchangeable in real analysis. ...

Completeness of C(K) under the sup norm

Let (K,d)(K,d)(K,d) be a compact metric space and let C(K,R)C(K,\mathbb{R})C(K,R) denote the set of continuous functions f:K→Rf:K\to\mathbb{R}f:K→R. Define the sup norm by ∥f∥∞=sup⁡x∈K∣f(x)∣, \|f\|_\infty=\sup_{x\in K}|f(x)|, ∥f∥∞=supx∈K∣f(x)∣, and the induced metric d∞(f,g)=∥f−g∥∞=sup⁡x∈K∣f(x)−g(x)∣. d_\infty(f,g)=\|f-g\|_\infty=\sup_{x\in K}|f(x)-g(x)|. d∞(f,g)=∥f−g∥∞=supx∈K∣f(x)−g(x)∣. ...

Complex conjugate

For z=a+bi∈Cz=a+bi\in\mathbb{C}z=a+bi∈C with a,b∈Ra,b\in\mathbb{R}a,b∈R, the complex conjugate of zzz is z‾:=a−bi.\overline{z}:=a-bi.z:=a−bi.Complex conjugation is an algebraic involution: it reverses the sign of the imaginary part and satisfies z‾‾=z\overline{\overline{z}}=zz=z. It is used to express the modulus and to compute inverses: if z≠0z\ne 0z=0 then z−1=z‾/(zz‾)z^{-1}=\overline{z}/(z\overline{z})z−1=z/(zz). ...

Complex numbers

The complex numbers are C:={a+bi:a,b∈R, i2=−1}.\mathbb{C}:=\{a+bi : a,b\in\mathbb{R},\ i^2=-1\}.C:={a+bi:a,b∈R, i2=−1}. Addition and multiplication are defined by (a+bi)+(c+di)=(a+c)+(b+d)i,(a+bi)+(c+di)=(a+c)+(b+d)i,(a+bi)+(c+di)=(a+c)+(b+d)i, (a+bi)(c+di)=(ac−bd)+(ad+bc)i.(a+bi)(c+di)=(ac-bd)+(ad+bc)i.(a+bi)(c+di)=(ac−bd)+(ad+bc)i.The field C\mathbb{C}C is a two-dimensional real vector space and can be identified with R2\mathbb{R}^2R2 via a+bi↔(a,b)a+bi\leftrightarrow(a,b)a+bi↔(a,b). Complex numbers are the natural setting for Fourier analysis, power series, and many aspects of analysis and geometry. ...

Composition of functions

Let f:X→Yf:X\to Yf:X→Y and g:Y→Zg:Y\to Zg:Y→Z be functions. The composition of ggg with fff is the function g∘f:X→Z,(g∘f)(x):=g(f(x)).g\circ f:X\to Z,\qquad (g\circ f)(x):=g(f(x)).g∘f:X→Z,(g∘f)(x):=g(f(x)).Composition formalizes successive application of maps and is the basic operation for building new functions from old ones. In analysis, the chain rule concerns derivatives of compositions, and continuity is stable under composition. ...

Composition preserves Riemann integrability

Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be Riemann integrable , and let g:R→Rg:\mathbb{R}\to\mathbb{R}g:R→R be continuous . Proposition: The composition g∘fg\circ fg∘f is Riemann integrable on [a,b][a,b][a,b]. More generally, it suffices that ggg be continuous on a closed interval containing the compact set f([a,b])f([a,b])f([a,b]) (since fff is bounded ). This proposition is used constantly to deduce integrability of ∣f∣|f|∣f∣, f2f^2f2, sin⁡(f)\sin(f)sin(f), etc., from integrability of fff. ...

Conditionally convergent series

A series ∑n=1∞an\sum_{n=1}^\infty a_n∑n=1∞an (with an∈Ra_n\in\mathbb{R}an∈R or C\mathbb{C}C) is conditionally convergent if: ∑n=1∞an\sum_{n=1}^\infty a_n∑n=1∞an converges, and ∑n=1∞∣an∣\sum_{n=1}^\infty |a_n|∑n=1∞∣an∣ diverges. Conditional convergence is a specifically infinite-dimensional phenomenon: it is responsible for rearrangement pathology (e.g., Riemann rearrangement theorem in R\mathbb{R}R). ...