# Analysis Definitions

 .remove_background_ad { border: 1px solid #555555; padding: .75em; margin: .75em; background-color: #e7e7e7; } .rmbg_image { max-height: 80px; } Isometric embedding A function g : (Y, p) → (X, d) that preserves the metric, meaning p(y, z) = d(g(y), g(z)) Isometry A surjective isometric embedding. A function g : (Y, p) → (X, d) that preserves the metric, meaning p(y, z) = d(g(y), g(z)) Converging sequence (xn) ⊆ X, with xn→ x ∈ X d(x, xn) → 0 Projection to the i-th coordinate πi : Rn → Rπi(x1, x2, . . . xi, . . . xn) = xi Cauchy For every ε > 0, there exists N ∈ N such that for all n, m > N, d(xn, xm) < ε Complete Every Cauchy sequence converges Banach space A complete normed linear space Hilbert space Complete inner product space Lipschitz function with constant c ≥ 0 d(f(y), f(z)) ≤ cp(y, z) for all y, z ∈ Y Contraction Lipschitz function with constant c < 1 Open metric ball about p of radius r in X B(p, r) = {x ∈ X : d(x, p) < r} Bounded set A ∈ X For some x ∈ X and some r > 0, A ⊆ B(x, r) Fixed point of f : X → X F(x) = x Fixed point property Any continuous function mapping that set to itself has a fixed point Interior of a set A ⊆ X Int A = {x ∈ X : ∃r > 0 with B(x, r) ⊆ A}Really inside A (have some metric ball contained in A) Exterior of a set A ⊆ X Ext A = {x ∈ X : ∃r > 0 with B(x, r) ∩ A = ∅}Really outside A (have some metric ball which misses A entirely) Boundary of a set A ⊆ X ∂A = {x ∈ X : ∀r > 0, B(x, r) ∩ A ≠ ∅ and B(x, r) − A ≠ ∅}Right on the edge (every metric ball is part inside and part outside) Open subset U ⊆ X U = Int UEvery point is an interior point Closed subset C ⊆ X ∂C ⊆ CContains all its boundary points Limit point b of C ⊆ X ∃ (cn) ⊆ C − {b} with cn → bThere is a sequence that converges to b Closure of A ⊆ X Ā = A ∪ ∂A A function ƒ is continuous at b ∈ X For any sequence (xn) ⊆ X with xn → b, ƒ(xn) → f(b) A function ƒ is continuous ƒ is continuous at each point of X Coordinate functions for a function F : X → Rm ƒ1, . . . ƒm : X → R defined by ƒi = πi∘F Open function ƒ : X → Y ∀ U open in X, ƒ(U) is open in YMaps open sets to open sets Homeomorphism A continuous, bijective function ƒ : X → Y such that ƒ-1 : Y → X is also continuousƒ preserves any properties which can be defined using open sets, such as convergence of sequencesEvery isometry is a homeomorphism, but not visa versa Embedding ƒ : X → Y such that ƒ : X → ƒ(X) is a homeomorphism, where ƒ(X) is a subspace of Y Compact X such that every sequence has a subsequence which converges to a point of X Distance between A, B ⊆ X d(A, B) = inf d(a, b) where a ∈ A, b ∈ B Maximizer of ƒ : A → R a ∈ A such that ƒ(a) ≥ ƒ(x) for all x ∈ A Strong maximizer of ƒ : A → R a ∈ A such that ƒ(a) > ƒ(x) for all x ∈ A - {a} Minimizer of ƒ : A → R b ∈ A such that ƒ(b) ≤ ƒ(x) for all x ∈ A Strong minimizer of ƒ : A → R b ∈ A such that ƒ(b) < ƒ(x) for all x ∈ A - {b} Extreme of ƒ : A → R A minimizer or maximizer of ƒ Uniformly continuous A function ƒ : X → Y such that for every ε > 0, there exists δ > 0 such that d(ƒ(u), ƒ(v)) < ε for any u, v ∈ X with d(u, v) < δ Separation of X (U, V), where U, V are open nonempty subsets of X such that X = U ∪ V and U ∩ V = ∅ Disconnected A metric space that has a separation (can be written as the union of nonempty open sets) Connected A metric space that is not disconnected (cannot be written as the union of nonempty open sets) Path in X A continuous function γ : [a, b] → X, where a, b ∈ R and a < bA path from γ(a) to γ(b) Path connected A metric space X such that for each x, y ∈ X, there is a path from x to y in X Convex A subset A of a normed linear space V such that for any v, u ∈ A, the line segment from u to v is in A limx→b ƒ(x) = L For every sequence (xn) ⊆ X − {b} with xn → b, ƒ(xn) → L, where b is a limit point of X ƒ approximates g to n-th order at b g(b) = ƒ(b) and limx→b d(ƒ(x), g(x))∕[d(x, b)]n = 0 Linear function L : V → W for V, W real vector spaces, such that L(αv) = αL(v) and L(v + u) = L(v) + L(u) for any v, u ∈ V and α ∈ RPreserves addition and scalar multiplication Affine function T : V → W defined by T(x) = L(x) + c, where L is linear and c ∈ WConstant distance from a linear function; only for real vector spaces Differentiable at b F : U → Rm where U is open in Rn, such that there is an affine function Tb which approximates F to 1-order at b Differentiable A function that is differentiable at all points in the domain Rn × m The collection of n × m matrices with real entries Norm on Rn × m ||A|| = √(∑i=1n ∑j=1m ai, j2) Gradient of ƒ at b ∈ U Cn(U, R) {ƒ : U → R : all partials of degree at most n exist and are continuous} where U is open in Rn ƒ ∈ C∞(U, R) A function that is Cn for all n Directional derivative For ƒ : U → R at b ∈ U (an open subset of Rn) with respect to u ∈ Rn Local maximizer of ƒ : X → R p ∈ X such that there is an open U∋p such that ƒ(p) ≥ ƒ(x) for all x ∈ U Strong local maximizer of ƒ : X → R p ∈ X such that there is an open U∋p such that ƒ(p) > ƒ(x) for all x ∈ U − {p} Local minimizer of ƒ : X → R p ∈ X such that there is an open U∋p such that ƒ(p) ≤ ƒ(x) for all x ∈ U Strong local minimizer of ƒ : X → R p ∈ X such that there is an open U∋p such that ƒ(p) < ƒ(x) for all x ∈ U − {p} Local extreme of ƒ : X → R A local minimizer or maximizer of ƒ dxi(v) vi, where v = (v1 . . . vn) an n-vector Differential 0-form on A ⊆ Rn A function ƒ : A → R Differential 1-form ω on A ⊆ Rn ω : A × Rn → RGiven coefficient functions fi : A → R for 1 ≤ i ≤ n, ω = ƒ1dx1 + . . . + ƒndxn Orientation on P, a polygon in R2 A consistent choice of direction on the boundary of P (CCW is positive, CW is negative) Oriented area of P Normal area of P times the sign of the orientation Oriented polygon in Rn T(P), where T : R2 → Rn is affine, and P is a polygon in R2 dxi ∧ dxj(P) for P an oriented polygon in Rn The oriented area of the projection Q of P into the xixj-plane Oriented area of a parallelogram P spanned by u, v n-vectors Differential 2-form Oriented r-volume of the projection Q of the parallelepiped spanned by u1, . . . ur in the xi1, . . . xir-hyperplane Differential r-form  Basic r-form Simple r-form ω a Cn form Each coefficient function of ω is Cn Exterior derivative of a differentiable 0-form ƒ : U → R Jacobian matrix of F at p ∈ U For F : U → Rm where U is open in Rn, with component functions ƒiaka total derivative Total derivative of F at p ∈ U For F : U → Rm where U is open in Rn, with component functions ƒiaka jacobian matrix Directional derivative of F at p ∈ U with respect to v ∈ Rn For F : U → Rm where U is open in Rn Smooth at p ∈ U F : U → Rm where U is open in Rn, such that DF(p) has rank nTotal derivative has rank (dimension of row space) equal to number of rows c-level hyper-surface Sc = {x ∈ U : ƒ(x) = c} for ƒ : U → R where U is open in Rn, and c ∈ R i-tensor on a vector space V over the real numbers A function Γ : Vi → R that is linear in each coordinate, i.e. for each 1 ≤ k ≤ i, Rn1× . . . × ni The set of i-dimensional rectangles of numbers with side lengths nk in the k-th direction ||A|| for A ∈ Rn1× . . . × ni The square root of the sum of the squares of the entries of AIsometric to Rn1 . . . ni, so it's a norm A(w)i m-th Taylor polynomial to ƒ at b ∈ U For ƒ ∈ Cm(U, R) where U is open in Rn, Positive definite tensor An i-tensor Γ on Rn such that for any non-zero u ∈ Rn, Γ(u, . . . u) > 0 Negative definite tensor An i-tensor Γ on Rn such that for any non-zero u ∈ Rn, Γ(u, . . . u) < 0 Indefinite tensor An i-tensor Γ on Rn such that for there exist u, v ∈ Rn with Γ(u, . . . u) < 0 and Γ(v, . . . v) > 0 Positive definite matrix A ∈ Rn × . . . × n such that ΓA is positive definite Negative definite matrix A ∈ Rn × . . . × n such that ΓA is negative definite Indefinite matrix A ∈ Rn × . . . × n such that ΓA is indefinite Interval of Rn I = I1 × . . . × In, where each Ii is an interval in R Closed interval of Rn Interval made up of only closed intervals Open interval of Rn Interval made up of only open intervals Volume of the interval I = I1 × . . . × In vol(I) = l(I1) . . . l(In), the product of the lengths of the intervals Partition of I = I1 × . . . × In Δ = Δ1 × . . . × Δn, where each Δi is a partition of Ii, a closed interval of R Subinterval J of Δ a partition of I The product of one sub-interval from each of the partitions that make up ΔWrite J ∈ Δ Maximum on a subinterval J ∈ Δ MJ = sup ƒ|J, where Δ is a partition of I, a closed interval of Rn, and ƒ : I → R a bounded function Minimum on a subinterval J ∈ Δ mJ = inf ƒ|J, where Δ is a partition of I, a closed interval of Rn, and ƒ : I → R a bounded function Upper sum of ƒ for the partition Δ Lower sum of ƒ for the partition Δ Refinement of the partition Δ = Δ1 × . . . × Δn Π = Π1 × . . . × Πn, where each Πi is a refinement of Δi (meaning Δi ⊆ Πi) Common refinement of the partitions Δ = Δ1 × . . . × Δn and Π = Π1 × . . . × Πn [Δ1 ∪ Π1] × . . . × [Δn ∪ Πn] A function ƒ : I → R is integrable, where I is a closed interval of Rn and ƒ is bounded sup{L(ƒ, Δ) : Δ is a partition of I} = inf{U(ƒ, Π) : Π is a partition of I} Integral of an integrable function ƒ Diameter of A For X a metric space, and A ⊆ X, Mesh of Δ n-cube A closed interval I = I1 × . . . × In of Rn with the property that l(I1) = . . . = l(In) Cubical partition of Δ Every subinterval of Δ is an n-cube; Extension by zero of ƒ to B For A ⊆ B ⊆ Rn and ƒ : A → R, g : B → R is the extension, defined by Characteristic function of the set A ⊆ Rn with A bounded A ⊆ Rn has well-defined volume χA is integrable; define Translate of A ⊆ Rn by u ∈ Rn A + u = {a + u : a ∈ A} Jordan domain A bounded subset D of Rn such that vol ∂D = 0 Parallelepiped based at p ∈ Rn spanned by n-vectors u1, . . . um Rotation by θ ∈ R Rθ : R2 → R2 the linear function with matrix Hyperplane in Rn A translation of a n − 1 dimensional sub(vector) space of RnThe solution sets of a single linear equation Shear in the i-th coordinate in Rn Linear function L : Rn → Rn such that L(ej) = ej for j ≠ i, and L(ei) • ei = 1. Shear in the first coordinate is represented by Uniformly differentiable Jacobian of F : U → Rn differentiable at u ∈ U, with U open in Rn JF(u) = det DF(u) .remove_background_ad { border: 1px solid #555555; padding: .75em; margin: .75em; background-color: #e7e7e7; } .rmbg_image { max-height: 80px; } Authormathprodigy20 ID54858 Card SetAnalysis Definitions DescriptionList of definitions for Math 342 Updated2010-12-14T14:25:21Z Show Answers