Divergent 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 divergent if its sequence of partial sums sN:=∑n=1Nans_N := \sum_{n=1}^N a_nsN:=n=1∑Nan does not converge in R\mathbb{R}R or C\mathbb{C}C. ...

Domain

If f:X→Yf:X\to Yf:X→Y is a function, then the domain of fff is the set XXX. The domain is part of the data of a function. In analysis, changing the domain can change key properties (e.g., injectivity of x↦x2x\mapsto x^2x↦x2 on R\mathbb{R}R versus on [0,∞)[0,\infty)[0,∞)). Examples: For f:R→Rf:\mathbb{R}\to\mathbb{R}f:R→R, f(x)=x2f(x)=x^2f(x)=x2, the domain is R\mathbb{R}R. For log⁡:(0,∞)→R\log:(0,\infty)\to\mathbb{R}log:(0,∞)→R, the domain is (0,∞)(0,\infty)(0,∞). The restriction f∣[0,1]f|_{[0,1]}f∣[0,1] has domain [0,1][0,1][0,1] even if fff was originally defined on R\mathbb{R}R.

Empty set

The empty set is the set ∅\varnothing∅ such that ∀x, (x∉∅).\forall x,\ (x\notin \varnothing).∀x, (x∈/∅).It is the identity for union in the sense that A∪∅=AA\cup \varnothing = AA∪∅=A, and it is the smallest set under ⊆\subseteq⊆. In analysis, it often appears as a preimage of an impossible condition (e.g., f−1(B)=∅f^{-1}(B)=\varnothingf−1(B)=∅ when BBB misses the range of fff). ...

Equicontinuity

Let (X,dX)(X,d_X)(X,dX) and (Y,dY)(Y,d_Y)(Y,dY) be metric spaces , and let F\mathcal{F}F be a family of functions f:X→Yf:X\to Yf:X→Y. The family F\mathcal{F}F is equicontinuous at a point x0∈Xx_0\in Xx0∈X if ∀ε>0, ∃δ>0 such that ∀f∈F, ∀x∈X, (dX(x,x0)<δ⇒dY(f(x),f(x0))<ε).\forall \varepsilon>0,\ \exists \delta>0\ \text{such that}\ \forall f\in\mathcal{F},\ \forall x\in X,\ \bigl(d_X(x,x_0)<\delta \Rightarrow d_Y(f(x),f(x_0))<\varepsilon\bigr).∀ε>0, ∃δ>0 such that ∀f∈F, ∀x∈X, (dX(x,x0)<δ⇒dY(f(x),f(x0))<ε). It is equicontinuous on XXX if it is equicontinuous at every x0∈Xx_0\in Xx0∈X. ...

Equicontinuity + pointwise boundedness implies uniform boundedness on compact sets

Let (K,d)(K,d)(K,d) be a compact metric space and let F⊆C(K,R)\mathcal{F}\subseteq C(K,\mathbb{R})F⊆C(K,R) be a family of continuous functions . Assume: F\mathcal{F}F is equicontinuous on KKK, and F\mathcal{F}F is pointwise bounded on KKK (for each x∈Kx\in Kx∈K, sup⁡f∈F∣f(x)∣<∞\sup_{f\in\mathcal{F}}|f(x)|<\inftysupf∈F∣f(x)∣<∞). Lemma: Then F\mathcal{F}F is uniformly bounded on KKK; i.e., there exists M<∞M<\inftyM<∞ such that ∣f(x)∣≤Mfor all f∈F, x∈K. |f(x)|\le M\quad \text{for all } f\in\mathcal{F},\ x\in K. ∣f(x)∣≤Mfor all f∈F, x∈K. ...

Equicontinuity plus dense-set convergence implies uniform convergence on compacta

Lemma (equicontinuity + dense-set convergence): Let KKK be a compact metric space and let (fn)(f_n)(fn) be a sequence of real-valued continuous functions on KKK that is equicontinuous : for every ε>0\varepsilon>0ε>0 there exists δ>0\delta>0δ>0 such that d(x,y)<δ ⟹ ∣fn(x)−fn(y)∣<εfor all n. d(x,y)<\delta \implies |f_n(x)-f_n(y)|<\varepsilon \quad \text{for all } n. d(x,y)<δ⟹∣fn(x)−fn(y)∣<εfor all n. Assume there exists a dense subset D⊆KD\subseteq KD⊆K such that for every x∈Dx\in Dx∈D, the sequence (fn(x))(f_n(x))(fn(x)) converges (equivalently, is Cauchy ). ...

Equicontinuous family

Let (X,dX)(X,d_X)(X,dX) and (Y,dY)(Y,d_Y)(Y,dY) be metric spaces , and let F\mathcal{F}F be a family of functions f:X→Yf:X\to Yf:X→Y. The family F\mathcal{F}F is equicontinuous at x0∈Xx_0\in Xx0∈X if ∀ε>0 ∃δ>0 ∀f∈F ∀x∈X: dX(x,x0)<δ ⟹ dY(f(x),f(x0))<ε. \forall \varepsilon>0\ \exists \delta>0\ \forall f\in\mathcal{F}\ \forall x\in X:\ d_X(x,x_0)<\delta \implies d_Y\bigl(f(x),f(x_0)\bigr)<\varepsilon. ∀ε>0 ∃δ>0 ∀f∈F ∀x∈X: dX(x,x0)<δ⟹dY(f(x),f(x0))<ε. The family F\mathcal{F}F is equicontinuous on XXX if it is equicontinuous at every x0∈Xx_0\in Xx0∈X (with δ\deltaδ allowed to depend on x0x_0x0 and ε\varepsilonε, but not on fff). ...

Equivalence class

Let ∼\sim∼ be an equivalence relation on a set XXX. For x∈Xx\in Xx∈X, the equivalence class of xxx is [x]:={y∈X:y∼x}⊆X.[x] := \{y\in X : y\sim x\}\subseteq X.[x]:={y∈X:y∼x}⊆X.Equivalence classes are the “blocks” determined by ∼\sim∼; they are pairwise disjoint and their union is all of XXX. Quotient constructions replace elements by their classes. Examples: If ∼\sim∼ is congruence mod 333 on Z\mathbb{Z}Z, then [1]={…,−5,−2,1,4,7,… }[1]=\{\dots,-5,-2,1,4,7,\dots\}[1]={…,−5,−2,1,4,7,…}. If ∼\sim∼ is equality on XXX, then [x]={x}[x]=\{x\}[x]={x} for each x∈Xx\in Xx∈X. On R\mathbb{R}R with x∼y ⟺ x−y∈Qx\sim y \iff x-y\in\mathbb{Q}x∼y⟺x−y∈Q, the class of 000 is [0]=Q[0]=\mathbb{Q}[0]=Q, and the class of 2\sqrt{2}2 is 2+Q\sqrt{2}+\mathbb{Q}2+Q.

Equivalence relation

An equivalence relation on a set XXX is a relation ∼ ⊆X×X\sim\ \subseteq X\times X∼ ⊆X×X such that for all x,y,z∈Xx,y,z\in Xx,y,z∈X: (Reflexive) x∼xx\sim xx∼x. (Symmetric) If x∼yx\sim yx∼y, then y∼xy\sim xy∼x. (Transitive) If x∼yx\sim yx∼y and y∼zy\sim zy∼z, then x∼zx\sim zx∼z. Equivalence relations formalize “sameness up to a criterion.” They partition XXX into equivalence classes , and many constructions in mathematics are quotients by equivalence relations. ...

Equivalent definitions of continuity (metric spaces)

Let (X,dX)(X,d_X)(X,dX) and (Y,dY)(Y,d_Y)(Y,dY) be metric spaces , and let f:X→Yf:X\to Yf:X→Y. Fix a point x0∈Xx_0\in Xx0∈X. The following are equivalent: Epsilon–delta continuity at x0x_0x0: for every ε>0\varepsilon>0ε>0 there exists δ>0\delta>0δ>0 such that dX(x,x0)<δ ⟹ dY(f(x),f(x0))<ε. d_X(x,x_0)<\delta \implies d_Y(f(x),f(x_0))<\varepsilon. dX(x,x0)<δ⟹dY(f(x),f(x0))<ε. Sequential continuity at x0x_0x0: for every sequence (xn)(x_n)(xn) in XXX, xn→x0 ⟹ f(xn)→f(x0). x_n\to x_0 \implies f(x_n)\to f(x_0). xn→x0⟹f(xn)→f(x0). Neighborhood formulation: for every open set V⊆YV\subseteq YV⊆Y with f(x0)∈Vf(x_0)\in Vf(x0)∈V, there exists δ>0\delta>0δ>0 such that BX(x0,δ)⊆f−1(V). B_X(x_0,\delta)\subseteq f^{-1}(V). BX(x0,δ)⊆f−1(V). Moreover, fff is continuous on all of XXX if and only if preimages of open sets are open: V⊆Y open ⟹ f−1(V)⊆X open. V\subseteq Y \text{ open} \implies f^{-1}(V)\subseteq X \text{ open}. V⊆Y open⟹f−1(V)⊆X open. ...