For x=(x1,…,xk)x=(x_1,\dots,x_k)x=(x1​,…,xk​) and y=(y1,…,yk)y=(y_1,\dots,y_k)y=(y1​,…,yk​) in Rk\mathbb{R}^kRk, the (standard) inner product is ⟨x,y⟩:=∑i=1kxiyi.\langle x,y\rangle := \sum_{i=1}^k x_i y_i.⟨x,y⟩:=i=1∑k​xi​yi​.The inner product is bilinear, symmetric, and positive definite, and it generates the Euclidean norm by ∥x∥2=⟨x,x⟩\|x\|_2=\sqrt{\langle x,x\rangle}∥x∥2​=⟨x,x⟩​. It is the algebraic structure behind orthogonality, projections, and many inequalities. ...

Integral Test: Let f:[1,∞)→Rf:[1,\infty)\to\mathbb{R}f:[1,∞)→R be positive, decreasing, and continuous, and set an=f(n)a_n=f(n)an​=f(n). Then ∑n=1∞an converges ⟺∫1∞f(x) dx converges.\sum_{n=1}^\infty a_n \text{ converges } \Longleftrightarrow \int_1^\infty f(x)\,dx \text{ converges}.∑n=1∞​an​ converges ⟺∫1∞​f(x)dx converges. ...

Let f,g:[a,b]→Rf,g:[a,b]\to\mathbb{R}f,g:[a,b]→R be continuously differentiable (i.e., f,g∈C1([a,b])f,g\in C^1([a,b])f,g∈C1([a,b])). Then f′f'f′, g′g'g′, fgfgfg are continuous and hence Riemann integrable . Corollary (integration by parts): $ \int_a^b f(x),g’(x),dx f(b)g(b)-f(a)g(a)-\int_a^b f’(x),g(x),dx. $ Integration by parts is a fundamental transformation tool in analysis, especially for estimating integrals and manipulating Fourier-type expressions. ...

Integration by parts (Riemann–Stieltjes): Let f,g:[a,b]→Rf,g:[a,b]\to\mathbb{R}f,g:[a,b]→R be functions of bounded variation, and assume at least one of fff or ggg is continuous on [a,b][a,b][a,b]. Then both Riemann–Stieltjes integrals ∫abf dg\int_a^b f\,dg∫ab​fdg and ∫abg df\int_a^b g\,df∫ab​gdf exist and ∫abf dg+∫abg df=f(b)g(b)−f(a)g(a). \int_a^b f\,dg + \int_a^b g\,df = f(b)g(b)-f(a)g(a). ∫ab​fdg+∫ab​gdf=f(b)g(b)−f(a)g(a). ...

In the Riemann–Stieltjes integral ∫abf dα\int_a^b f\,d\alpha∫ab​fdα, the function α:[a,b]→R\alpha:[a,b]\to\mathbb{R}α:[a,b]→R is called the integrator (or integrator function). For a partition P:a=x0<⋯<xn=bP:a=x_0<\cdots<x_n=bP:a=x0​<⋯<xn​=b, the weights are the increments Δαi=α(xi)−α(xi−1).\Delta\alpha_i=\alpha(x_i)-\alpha(x_{i-1}).Δαi​=α(xi​)−α(xi−1​).Typically one assumes α\alphaα is increasing (or more generally of bounded variation) to ensure good behavior of the integral and to guarantee that upper/lower sum definitions make sense. ...

Let fn:[a,b]→Rf_n:[a,b]\to\mathbb{R}fn​:[a,b]→R be Riemann integrable for each nnn, and suppose fn→ff_n\to ffn​→f uniformly on [a,b][a,b][a,b]. Corollary: The limit function fff is Riemann integrable, and one may interchange limit and integral: lim⁡n→∞∫abfn(x) dx=∫ablim⁡n→∞fn(x) dx. \lim_{n\to\infty}\int_a^b f_n(x)\,dx=\int_a^b \lim_{n\to\infty} f_n(x)\,dx. limn→∞​∫ab​fn​(x)dx=∫ab​limn→∞​fn​(x)dx. Connection to parent theorem: This is the “uniform convergence and integration ” theorem (often stated in exactly this corollary form). Uniform boundedness is automatic under uniform convergence and boundedness of some fNf_NfN​. ...

Let (X,d)(X,d)(X,d) be a metric space and let A⊆XA\subseteq XA⊆X. The interior of AAA, denoted int⁡(A)\operatorname{int}(A)int(A) (or A∘A^\circA∘), is the set int⁡(A):={x∈A:∃r>0 with B(x,r)⊆A}\operatorname{int}(A):=\{x\in A : \exists r>0\ \text{with}\ B(x,r)\subseteq A\}int(A):={x∈A:∃r>0 with B(x,r)⊆A} (see open ball ). Equivalently, int⁡(A)\operatorname{int}(A)int(A) is the union of all open sets contained in AAA. The interior captures the points of AAA that are not “on the edge” (compare with boundary ). ...

Intermediate Value Theorem: Let I⊆RI\subseteq\mathbb{R}I⊆R be an interval and let f:I→Rf:I\to\mathbb{R}f:I→R be continuous . If a,b∈Ia,b\in Ia,b∈I with a<ba<ba<b and yyy is any real number between f(a)f(a)f(a) and f(b)f(b)f(b), then there exists c∈[a,b]c\in[a,b]c∈[a,b] such that f(c)=y.f(c)=y.f(c)=y. This theorem formalizes the idea that continuous functions on intervals cannot “jump over” values, and it is the basis for existence of roots, fixed points on intervals, and many approximation results. ...

The intersection of sets AAA and BBB is A∩B:={x:(x∈A) ∧ (x∈B)}.A\cap B := \{x : (x\in A)\ \land\ (x\in B)\}.A∩B:={x:(x∈A) ∧ (x∈B)}. More generally, for an indexed family {Ai}i∈I\{A_i\}_{i\in I}{Ai​}i∈I​, the intersection is ⋂i∈IAi:={x:∀i∈I, x∈Ai}.\bigcap_{i\in I} A_i := \{x : \forall i\in I,\ x\in A_i\}.i∈I⋂​Ai​:={x:∀i∈I, x∈Ai​}.Intersections encode simultaneous constraints. In topology, closed sets are closed under arbitrary intersections, and limit-point definitions often involve intersections of neighborhoods. ...

A subset I⊆RI\subseteq\mathbb{R}I⊆R is an interval if ∀x,y∈I, ∀t∈R, (x<t<y⇒t∈I),\forall x,y\in I,\ \forall t\in\mathbb{R},\ \bigl(x<t<y \Rightarrow t\in I\bigr),∀x,y∈I, ∀t∈R, (x<t<y⇒t∈I), i.e. whenever x,y∈Ix,y\in Ix,y∈I and x<yx<yx<y, every real number between them is also in III. Intervals are exactly the connected subsets of R\mathbb{R}R (with the usual topology), and they form the natural domains for one-variable analysis (limits, derivatives, integrals). ...