Analysis Definitions

A function g : (Y, p) → (X, d) that preserves the metric, meaning p(y, z) = d(g(y), g(z))

• A surjective isometric embedding.
• A function g : (Y, p) → (X, d) that preserves the metric, meaning p(y, z) = d(g(y), g(z))

d(x, x_n) → 0

• π_i : Rⁿ → R
• π_i(x₁, x₂, . . . x_i, . . . x_n) = x_i

For every ε > 0, there exists N ∈ N such that for all n, m > N, d(x_n, x_m) < ε

Every Cauchy sequence converges

A complete normed linear space

Complete inner product space

d(f(y), f(z)) ≤ cp(y, z) for all y, z ∈ Y

Lipschitz function with constant c < 1

B(p, r) = {x ∈ X : d(x, p) < r}

For some x ∈ X and some r > 0, A ⊆ B(x, r)

F(x) = x

Any continuous function mapping that set to itself has a fixed point

• Int A = {x ∈ X : ∃r > 0 with B(x, r) ⊆ A}
• Really inside A (have some metric ball contained in A)

• Ext A = {x ∈ X : ∃r > 0 with B(x, r) ∩ A = ∅}
• Really outside A (have some metric ball which misses A entirely)

• ∂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)

• U = Int U
• Every point is an interior point

• ∂C ⊆ C
• Contains all its boundary points

• ∃ (c_n) ⊆ C − {b} with c_n → b
• There is a sequence that converges to b

Ā = A ∪ ∂A

For any sequence (x_n) ⊆ X with x_n → b, ƒ(x_n) → f(b)

ƒ is continuous at each point of X

ƒ₁, . . . ƒ_m : X → R defined by ƒ_i = π_i∘F

• ∀ U open in X, ƒ(U) is open in Y
• Maps open sets to open sets

• A continuous, bijective function ƒ : X → Y such that ƒ^-1 : Y → X is also continuous
• ƒ preserves any properties which can be defined using open sets, such as convergence of sequences
• Every isometry is a homeomorphism, but not visa versa

ƒ : X → Y such that ƒ : X → ƒ(X) is a homeomorphism, where ƒ(X) is a subspace of Y

X such that every sequence has a subsequence which converges to a point of X

d(A, B) = inf d(a, b) where a ∈ A, b ∈ B

a ∈ A such that ƒ(a) ≥ ƒ(x) for all x ∈ A

a ∈ A such that ƒ(a) > ƒ(x) for all x ∈ A - {a}

b ∈ A such that ƒ(b) ≤ ƒ(x) for all x ∈ A

b ∈ A such that ƒ(b) < ƒ(x) for all x ∈ A - {b}

A minimizer or maximizer of ƒ

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

(U, V), where U, V are open nonempty subsets of X such that X = U ∪ V and U ∩ V = ∅

A metric space that has a separation (can be written as the union of nonempty open sets)

A metric space that is not disconnected (cannot be written as the union of nonempty open sets)

• A continuous function γ : [a, b] → X, where a, b ∈ R and a < b
• A path from γ(a) to γ(b)

A metric space X such that for each x, y ∈ X, there is a path from x to y in X

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

For every sequence (x_n) ⊆ X − {b} with x_n → b, ƒ(x_n) → L, where b is a limit point of X

g(b) = ƒ(b) and lim_x→b d(ƒ(x), g(x))∕[d(x, b)]ⁿ = 0

• 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 ∈ R
• Preserves addition and scalar multiplication

• T : V → W defined by T(x) = L(x) + c, where L is linear and c ∈ W
• Constant distance from a linear function; only for real vector spaces

F : U → R^m where U is open in Rⁿ, such that there is an affine function T_b which approximates F to 1-order at b

A function that is differentiable at all points in the domain

The collection of n × m matrices with real entries

||A|| = √(∑_i=1ⁿ ∑_j=1^m a_{i, j}²)

{ƒ : U → R : all partials of degree at most n exist and are continuous} where U is open in Rⁿ

A function that is Cⁿ for all n

• For ƒ : U → R at b ∈ U (an open subset of Rⁿ) with respect to u ∈ Rⁿ
• 

p ∈ X such that there is an open U∋p such that ƒ(p) ≥ ƒ(x) for all x ∈ U

p ∈ X such that there is an open U∋p such that ƒ(p) > ƒ(x) for all x ∈ U − {p}

p ∈ X such that there is an open U∋p such that ƒ(p) ≤ ƒ(x) for all x ∈ U

p ∈ X such that there is an open U∋p such that ƒ(p) < ƒ(x) for all x ∈ U − {p}

A local minimizer or maximizer of ƒ

v_i, where v = (v₁ . . . v_n) an n-vector

A function ƒ : A → R

• ω : A × Rⁿ → R
• Given coefficient functions f_i : A → R for 1 ≤ i ≤ n, ω = ƒ₁dx₁ + . . . + ƒ_ndx_n

A consistent choice of direction on the boundary of P (CCW is positive, CW is negative)

Normal area of P times the sign of the orientation

T(P), where T : R² → Rⁿ is affine, and P is a polygon in R²

The oriented area of the projection Q of P into the x_ix_j-plane

• 

Each coefficient function of ω is Cⁿ

• For F : U → R^m where U is open in Rⁿ, with component functions ƒ_i
• aka total derivative
• 

• For F : U → R^m where U is open in Rⁿ, with component functions ƒ_i
• aka jacobian matrix
• 

• For F : U → R^m where U is open in Rⁿ
• 

• F : U → R^m where U is open in Rⁿ, such that DF(p) has rank n
• Total derivative has rank (dimension of row space) equal to number of rows

S_c = {x ∈ U : ƒ(x) = c} for ƒ : U → R where U is open in Rⁿ, and c ∈ R

• A function Γ : Vⁱ → R that is linear in each coordinate, i.e. for each 1 ≤ k ≤ i,
• 

The set of i-dimensional rectangles of numbers with side lengths n_k in the k-th direction

• The square root of the sum of the squares of the entries of A
• Isometric to R^{n₁ . . . n_i}, so it's a norm

• For ƒ ∈ C^m(U, R) where U is open in Rⁿ,
• 

An i-tensor Γ on Rⁿ such that for any non-zero u ∈ Rⁿ, Γ(u, . . . u) > 0

An i-tensor Γ on Rⁿ such that for any non-zero u ∈ Rⁿ, Γ(u, . . . u) < 0

An i-tensor Γ on Rⁿ such that for there exist u, v ∈ Rⁿ with Γ(u, . . . u) < 0 and Γ(v, . . . v) > 0

A ∈ R^{n × . . . × n} such that Γ_A is positive definite

A ∈ R^{n × . . . × n} such that Γ_A is negative definite

A ∈ R^{n × . . . × n} such that Γ_A is indefinite

I = I₁ × . . . × I_n, where each I_i is an interval in R

Interval made up of only closed intervals

Interval made up of only open intervals

vol(I) = l(I₁) . . . l(I_n), the product of the lengths of the intervals

Δ = Δ₁ × . . . × Δ_n, where each Δ_i is a partition of I_i, a closed interval of R

• The product of one sub-interval from each of the partitions that make up Δ
• Write J ∈ Δ

M_J = sup ƒ|_J, where Δ is a partition of I, a closed interval of Rⁿ, and ƒ : I → R a bounded function

m_J = inf ƒ|_J, where Δ is a partition of I, a closed interval of Rⁿ, and ƒ : I → R a bounded function

Π = Π₁ × . . . × Π_n, where each Π_i is a refinement of Δ_i (meaning Δ_i ⊆ Π_i)

[Δ₁ ∪ Π₁] × . . . × [Δ_n ∪ Π_n]

sup{L(ƒ, Δ) : Δ is a partition of I} = inf{U(ƒ, Π) : Π is a partition of I}

• For X a metric space, and A ⊆ X,
• 

A closed interval I = I₁ × . . . × I_n of Rⁿ with the property that l(I₁) = . . . = l(I_n)

Every subinterval of Δ is an n-cube;

• For A ⊆ B ⊆ Rⁿ and ƒ : A → R, g : B → R is the extension, defined by
• 

• χ_A is integrable; define
• 

A + u = {a + u : a ∈ A}

A bounded subset D of Rⁿ such that vol ∂D = 0

• R_θ : R² → R² the linear function with matrix
• 

• A translation of a n − 1 dimensional sub(vector) space of Rⁿ
• The solution sets of a single linear equation

• Linear function L : Rⁿ → Rⁿ such that L(e_j) = e_j for j ≠ i, and L(e_i) • e_i = 1. Shear in the first coordinate is represented by
• 

J_F(u) = det DF(u)