Let (an)n∈N(a_n)_{n\in\mathbb{N}}(an​)n∈N​ be a sequence in R\mathbb{R}R or C\mathbb{C}C. The series ∑n=1∞an\sum_{n=1}^\infty a_nn=1∑∞​an​ is defined via its partial sums sN=∑n=1Nans_N=\sum_{n=1}^N a_nsN​=∑n=1N​an​. One says the series is summable if the sequence (sN)(s_N)(sN​) converges . ...

Let XXX be a set and let (Y,dY)(Y,d_Y)(Y,dY​) be a metric space (typically Y=RY=\mathbb{R}Y=R or C\mathbb{C}C). A series of functions on XXX is a formal expression ∑n=1∞fn,\sum_{n=1}^\infty f_n,n=1∑∞​fn​, where each fn:X→Yf_n:X\to Yfn​:X→Y. The associated partial sums are the functions ...

A set is an object AAA for which it makes sense to ask, for any object xxx, whether xxx is an element of AAA, written x∈Ax \in Ax∈A. In rigorous foundations (e.g., ZFC set theory), “set” and the membership relation ∈\in∈ are taken as primitive notions satisfying axioms. In analysis, one typically uses sets to collect numbers, points, functions , or other mathematical objects into a single entity that can be quantified over. Key operations include union , intersection , and subset relations. ...

The set difference (or relative complement) of BBB in AAA is A∖B:={x∈A:x∉B}.A\setminus B := \{x\in A : x\notin B\}.A∖B:={x∈A:x∈/B}.Set difference formalizes “AAA with the elements of BBB removed.” In analysis it appears in punctured neighborhoods (e.g., B(x,r)∖{x}B(x,r)\setminus\{x\}B(x,r)∖{x}) and in decompositions like (A∪B)∖A=B∖A(A\cup B)\setminus A = B\setminus A(A∪B)∖A=B∖A. Examples: {1,2,3}∖{2}={1,3}\{1,2,3\}\setminus\{2\}=\{1,3\}{1,2,3}∖{2}={1,3}. (0,2)∖(1,3)=(0,1](0,2)\setminus(1,3)=(0,1](0,2)∖(1,3)=(0,1]. For any set AAA, A∖∅=AA\setminus\varnothing = AA∖∅=A and A∖A=∅A\setminus A=\varnothingA∖A=∅.

A set N⊆RkN\subseteq\mathbb{R}^kN⊆Rk has (Lebesgue) measure zero (or is a null set) if for every ε>0\varepsilon>0ε>0 there exists a countable collection of kkk-dimensional rectangles (boxes) {Rn}n=1∞\{R_n\}_{n=1}^\infty{Rn​}n=1∞​ such that N⊆⋃n=1∞Rnand∑n=1∞vol⁡(Rn)<ε,N \subseteq \bigcup_{n=1}^\infty R_n \quad\text{and}\quad \sum_{n=1}^\infty \operatorname{vol}(R_n) < \varepsilon,N⊆n=1⋃∞​Rn​andn=1∑∞​vol(Rn​)<ε, where for a rectangle R=∏j=1k[aj,bj]R=\prod_{j=1}^k [a_j,b_j]R=∏j=1k​[aj​,bj​] its volume is ...

Let (X,d)(X,d)(X,d) be a metric space, let x∈Xx\in Xx∈X, and let r≥0r\ge 0r≥0. The (metric) sphere of radius rrr centered at xxx is S(x,r):={y∈X:d(x,y)=r}.S(x,r):=\{y\in X : d(x,y)=r\}.S(x,r):={y∈X:d(x,y)=r}.Spheres generalize the usual circles and spheres in Euclidean geometry. They are useful for describing boundaries of balls and for constructing examples in metric topology. Examples: In R\mathbb{R}R, S(a,r)={a−r,a+r}S(a,r)=\{a-r,a+r\}S(a,r)={a−r,a+r} if r>0r>0r>0. In R2\mathbb{R}^2R2, S(0,1)={(x,y):x2+y2=1}S(0,1)=\{(x,y):x^2+y^2=1\}S(0,1)={(x,y):x2+y2=1} is the unit circle. In a discrete metric space, S(x,1)=X∖{x}S(x,1)=X\setminus\{x\}S(x,1)=X∖{x}.

Squeeze Theorem (sequences): If an≤bn≤cna_n\le b_n\le c_nan​≤bn​≤cn​ for all sufficiently large nnn and an→L,cn→L,a_n\to L,\quad c_n\to L,an​→L,cn​→L, then bn→Lb_n\to Lbn​→L. Squeeze Theorem (functions): If g(x)≤f(x)≤h(x)g(x)\le f(x)\le h(x)g(x)≤f(x)≤h(x) near aaa (or for large xxx) and lim⁡x→ag(x)=lim⁡x→ah(x)=L\lim_{x\to a} g(x)=\lim_{x\to a} h(x)=Llimx→a​g(x)=limx→a​h(x)=L, then lim⁡x→af(x)=L\lim_{x\to a} f(x)=Llimx→a​f(x)=L. ...

Let [a,b]⊆R[a,b]\subseteq\mathbb{R}[a,b]⊆R and let PPP be a partition a=x0<x1<⋯<xn=ba=x_0<x_1<\cdots<x_n=ba=x0​<x1​<⋯<xn​=b. A function φ:[a,b]→R\varphi:[a,b]\to\mathbb{R}φ:[a,b]→R is a step function (with respect to PPP) if for each i=1,…,ni=1,\dots,ni=1,…,n there exists a constant ci∈Rc_i\in\mathbb{R}ci​∈R such that ...

Stone–Weierstrass Theorem (real version): Let KKK be a compact metric space and let A⊆C(K,R)A\subseteq C(K,\mathbb{R})A⊆C(K,R) be a subalgebra (closed under addition, multiplication, and scalar multiplication). Assume: AAA contains the constant functions, and AAA separates points: for any distinct x,y∈Kx,y\in Kx,y∈K there exists f∈Af\in Af∈A such that f(x)≠f(y)f(x)\neq f(y)f(x)=f(y). Then AAA is dense in C(K,R)C(K,\mathbb{R})C(K,R) with respect to the sup norm ∥⋅∥∞\|\cdot\|_\infty∥⋅∥∞​; i.e., for every g∈C(K,R)g\in C(K,\mathbb{R})g∈C(K,R) and ε>0\varepsilon>0ε>0 there exists f∈Af\in Af∈A with ∥g−f∥∞<ε. \|g-f\|_\infty<\varepsilon. ∥g−f∥∞​<ε. (For the complex version, one typically also assumes AAA is closed under complex conjugation.) ...

Let (xn)n∈N(x_n)_{n\in\mathbb{N}}(xn​)n∈N​ be a sequence in a set XXX. A subsequence of (xn)(x_n)(xn​) is a sequence of the form (xnk)k∈N(x_{n_k})_{k\in\mathbb{N}}(xnk​​)k∈N​, where (nk)k∈N(n_k)_{k\in\mathbb{N}}(nk​)k∈N​ is a strictly increasing sequence of natural numbers: ...