Residual set

Let (X,d)(X,d)(X,d) be a metric space . A set R⊆XR\subseteq XR⊆X is residual (or comeager) if its complement is meager : X∖R is meager in X. X\setminus R\ \text{is meager in }X. X∖R is meager in X. Residual sets are “topologically large” in complete metric spaces : by the Baire category theorem (see Baire space ), residual sets are dense (in fact, they contain a dense GδG_\deltaGδ set, though that terminology requires additional definitions). ...

Restriction of a function

Let f:X→Yf:X\to Yf:X→Y be a function and let A⊆XA\subseteq XA⊆X. The restriction of fff to AAA is the function f∣A:A→Y,f∣A(a):=f(a) for all a∈A.f|_A:A\to Y,\qquad f|_A(a):=f(a)\ \text{for all }a\in A.f∣A:A→Y,f∣A(a):=f(a) for all a∈A.Restrictions are used constantly in analysis to localize statements (e.g., continuity on a subset, behavior near a point, or defining inverses on domains where a function becomes injective). ...

Reverse triangle inequality

Reverse triangle inequality: In a normed vector space (V,∥⋅∥)(V,\|\cdot\|)(V,∥⋅∥), for all u,v∈Vu,v\in Vu,v∈V, ∣∥u∥−∥v∥∣≤∥u−v∥. \bigl|\|u\|-\|v\|\bigr|\le \|u-v\|. ∥u∥−∥v∥≤∥u−v∥. Equivalently, ∥u∥≤∥v∥+∥u−v∥and∥v∥≤∥u∥+∥u−v∥. \|u\|\le \|v\|+\|u-v\|\quad\text{and}\quad \|v\|\le \|u\|+\|u-v\|. ∥u∥≤∥v∥+∥u−v∥and∥v∥≤∥u∥+∥u−v∥. ...

Riemann integrability implies boundedness

Proposition: If f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R is Riemann integrable on [a,b][a,b][a,b], then fff is bounded on [a,b][a,b][a,b]. In many texts, boundedness is built into the definition of Riemann integrability. This proposition records that boundedness is not optional: unbounded functions cannot have finite upper and lower sums . Proof sketch: If fff were unbounded above, then for every partition PPP there would be some subinterval on which sup⁡f=+∞\sup f=+\inftysupf=+∞, forcing U(f,P)=+∞U(f,P)=+\inftyU(f,P)=+∞. Similarly if unbounded below, some lower sum would be −∞-\infty−∞. Thus the equality of upper and lower integrals (and finiteness of the integral) cannot hold unless fff is bounded. ...

Riemann integrable function

A bounded function f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R is Riemann integrable on [a,b][a,b][a,b] if ∀ε>0, ∃ a HAHAHUGOSHORTCODE782s1HBHB P of [a,b] such that U(f,P)−L(f,P)<ε.\forall \varepsilon>0,\ \exists\ \text{a }partition \ P\ \text{of }[a,b]\ \text{such that}\ U(f,P)-L(f,P)<\varepsilon.∀ε>0, ∃ a HAHAHUGOSHORTCODE782s1HBHB P of [a,b] such that U(f,P)−L(f,P)<ε.Equivalently, fff is Riemann integrable iff its upper integral inf⁡PU(f,P)\inf_P U(f,P)infPU(f,P) (using upper sums ) equals its lower integral sup⁡PL(f,P)\sup_P L(f,P)supPL(f,P) (using lower sums ), where the infimum/supremum are taken over all partitions PPP. ...

Riemann integral

If f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R is Riemann integrable , its Riemann integral over [a,b][a,b][a,b] is the common value ∫abf(x) dx:=sup⁡PL(f,P) = inf⁡PU(f,P),\int_a^b f(x)\,dx := \sup_P L(f,P) \;=\; \inf_P U(f,P),∫abf(x)dx:=PsupL(f,P)=PinfU(f,P), where PPP ranges over all partitions of [a,b][a,b][a,b], and L(f,P)L(f,P)L(f,P), U(f,P)U(f,P)U(f,P) are the lower and upper sums . ...

Riemann rearrangement theorem

Riemann rearrangement theorem: Let ∑n=1∞an\sum_{n=1}^\infty a_n∑n=1∞an be a conditionally convergent series of real numbers (i.e., it converges but not absolutely). Then for any L∈RL\in\mathbb{R}L∈R there exists a rearrangement ∑aπ(n)\sum a_{\pi(n)}∑aπ(n) that converges to LLL. There also exist rearrangements that diverge to +∞+\infty+∞, to −∞-\infty−∞, and that oscillate. ...

Riemann sum

Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be bounded and let (P,T)(P,T)(P,T) be a tagged partition with P:a=x0<⋯<xn=bP:a=x_0<\cdots<x_n=bP:a=x0<⋯<xn=b and tags ti∈[xi−1,xi]t_i\in[x_{i-1},x_i]ti∈[xi−1,xi]. The Riemann sum of fff for (P,T)(P,T)(P,T) is ...

Riemann–Stieltjes integrability theorem

Riemann–Stieltjes integrability theorem: Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be continuous , and let α:[a,b]→R\alpha:[a,b]\to\mathbb{R}α:[a,b]→R be increasing. Then the Riemann–Stieltjes integral ∫abf dα \int_a^b f\,d\alpha ∫abfdα exists. The Riemann–Stieltjes integral generalizes the Riemann integral (take α(x)=x\alpha(x)=xα(x)=x) and also encodes weighted sums (step-function α\alphaα) and distribution-function-type integrators . It is a standard bridge toward measure-theoretic integration. ...

Riemann–Stieltjes integral

Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be bounded, and let α:[a,b]→R\alpha:[a,b]\to\mathbb{R}α:[a,b]→R be a function. For a partition P:a=x0<⋯<xn=bP:a=x_0<\cdots<x_n=bP:a=x0<⋯<xn=b, define Δαi:=α(xi)−α(xi−1).\Delta\alpha_i:=\alpha(x_i)-\alpha(x_{i-1}).Δαi:=α(xi)−α(xi−1). For each iii, set ...