# Analysis Definitions

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1. Isometric embedding
A function g : (Y, p) → (X, d) that preserves the metric, meaning p(y, z) = d(g(y), g(z))
2. 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))
3. Converging sequence (xn) ⊆ X, with xn→ x ∈ X
d(x, xn) → 0
4. Projection to the i-th coordinate
• πi : Rn → R
• πi(x1, x2, . . . xi, . . . xn) = xi
5. Cauchy
For every ε > 0, there exists N ∈ N such that for all n, m > N, d(xn, xm) < ε
6. Complete
Every Cauchy sequence converges
7. Banach space
A complete normed linear space
8. Hilbert space
Complete inner product space
9. Lipschitz function with constant c ≥ 0
d(f(y), f(z)) ≤ cp(y, z) for all y, z ∈ Y
10. Contraction
Lipschitz function with constant c < 1
B(p, r) = {x ∈ X : d(x, p) < r}
12. Bounded set A ∈ X
For some x ∈ X and some r > 0, A ⊆ B(x, r)
13. Fixed point of f : X → X
F(x) = x
14. Fixed point property
Any continuous function mapping that set to itself has a fixed point
15. 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)
16. 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)
17. 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)
18. Open subset U ⊆ X
• U = Int U
• Every point is an interior point
19. Closed subset C ⊆ X
• ∂C ⊆ C
• Contains all its boundary points
20. Limit point b of C ⊆ X
• ∃ (cn) ⊆ C − {b} with cn → b
• There is a sequence that converges to b
21. Closure of A ⊆ X
Ā = A ∪ ∂A
22. A function ƒ is continuous at b ∈ X
For any sequence (xn) ⊆ X with xn → b, ƒ(xn) → f(b)
23. A function ƒ is continuous
ƒ is continuous at each point of X
24. Coordinate functions for a function F : X → Rm
ƒ1, . . . ƒm : X → R defined by ƒi = πi∘F
25. Open function ƒ : X → Y
• ∀ U open in X, ƒ(U) is open in Y
• Maps open sets to open sets
26. Homeomorphism
• 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
27. Embedding
ƒ : X → Y such that ƒ : X → ƒ(X) is a homeomorphism, where ƒ(X) is a subspace of Y
28. Compact
X such that every sequence has a subsequence which converges to a point of X
29. Distance between A, B ⊆ X
d(A, B) = inf d(a, b) where a ∈ A, b ∈ B
30. Maximizer of ƒ : A → R
a ∈ A such that ƒ(a) ≥ ƒ(x) for all x ∈ A
31. Strong maximizer of ƒ : A → R
a ∈ A such that ƒ(a) > ƒ(x) for all x ∈ A - {a}
32. Minimizer of ƒ : A → R
b ∈ A such that ƒ(b) ≤ ƒ(x) for all x ∈ A
33. Strong minimizer of ƒ : A → R
b ∈ A such that ƒ(b) < ƒ(x) for all x ∈ A - {b}
34. Extreme of ƒ : A → R
A minimizer or maximizer of ƒ
35. 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
36. Separation of X
(U, V), where U, V are open nonempty subsets of X such that X = U ∪ V and U ∩ V = ∅
37. Disconnected
A metric space that has a separation (can be written as the union of nonempty open sets)
38. Connected
A metric space that is not disconnected (cannot be written as the union of nonempty open sets)
39. Path in X
• A continuous function γ : [a, b] → X, where a, b ∈ R and a < b
• A path from γ(a) to γ(b)
40. Path connected
A metric space X such that for each x, y ∈ X, there is a path from x to y in X
41. 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
42. 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
43. 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
44. 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 ∈ R
• Preserves addition and scalar multiplication
45. Affine function
• 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
46. 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
47. Differentiable
A function that is differentiable at all points in the domain
48. Rn × m
The collection of n × m matrices with real entries
49. Norm on Rn × m
||A|| = √(∑i=1nj=1m ai, j2)
50. Gradient of ƒ at b ∈ U 51. Cn(U, R)
{ƒ : U → R : all partials of degree at most n exist and are continuous} where U is open in Rn
52. ƒ ∈ C(U, R)
A function that is Cn for all n
53. Directional derivative
• For ƒ : U → R at b ∈ U (an open subset of Rn) with respect to u ∈ Rn
• 54. 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
55. 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}
56. 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
57. 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}
58. Local extreme of ƒ : X → R
A local minimizer or maximizer of ƒ
59. dxi(v)
vi, where v = (v1 . . . vn) an n-vector
60. Differential 0-form on A ⊆ Rn
A function ƒ : A → R
61. Differential 1-form ω on A ⊆ Rn
• ω : A × Rn → R
• Given coefficient functions fi : A → R for 1 ≤ i ≤ n, ω = ƒ1dx1 + . . . + ƒndxn
62. Orientation on P, a polygon in R2
A consistent choice of direction on the boundary of P (CCW is positive, CW is negative)
63. Oriented area of P
Normal area of P times the sign of the orientation
64. Oriented polygon in Rn
T(P), where T : R2 → Rn is affine, and P is a polygon in R2
65. dxi ∧ dxj(P) for P an oriented polygon in Rn
The oriented area of the projection Q of P into the xixj-plane
66. Oriented area of a parallelogram P spanned by u, v n-vectors
• 67. Differential 2-form 68. Oriented r-volume of the projection Q of the parallelepiped spanned by u1, . . . ur in the xi1, . . . xir-hyperplane 69. Differential r-form  70. Basic r-form 71. Simple r-form 72. ω a Cn form
Each coefficient function of ω is Cn
73. Exterior derivative of a differentiable 0-form ƒ : U → R 74. Jacobian matrix of F at p ∈ U
• For F : U → Rm where U is open in Rn, with component functions ƒi
• aka total derivative
• 75. Total derivative of F at p ∈ U
• For F : U → Rm where U is open in Rn, with component functions ƒi
• aka jacobian matrix
• 76. Directional derivative of F at p ∈ U with respect to v ∈ Rn
• For F : U → Rm where U is open in Rn
• 77. Smooth at p ∈ U
• F : U → Rm 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
78. c-level hyper-surface
Sc = {x ∈ U : ƒ(x) = c} for ƒ : U → R where U is open in Rn, and c ∈ R
79. 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,
• 80. Rn1× . . . × ni
The set of i-dimensional rectangles of numbers with side lengths nk in the k-th direction
81. ||A|| for A ∈ Rn1× . . . × ni
• The square root of the sum of the squares of the entries of A
• Isometric to Rn1 . . . ni, so it's a norm
82. A(w)i 83. m-th Taylor polynomial to ƒ at b ∈ U
• For ƒ ∈ Cm(U, R) where U is open in Rn,
• 84. Positive definite tensor
An i-tensor Γ on Rn such that for any non-zero u ∈ Rn, Γ(u, . . . u) > 0
85. Negative definite tensor
An i-tensor Γ on Rn such that for any non-zero u ∈ Rn, Γ(u, . . . u) < 0
86. 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
87. Positive definite matrix
A ∈ Rn × . . . × n such that ΓA is positive definite
88. Negative definite matrix
A ∈ Rn × . . . × n such that ΓA is negative definite
89. Indefinite matrix
A ∈ Rn × . . . × n such that ΓA is indefinite
90. Interval of Rn
I = I1 × . . . × In, where each Ii is an interval in R
91. Closed interval of Rn
Interval made up of only closed intervals
92. Open interval of Rn
Interval made up of only open intervals
93. Volume of the interval I = I1 × . . . × In
vol(I) = l(I1) . . . l(In), the product of the lengths of the intervals
94. Partition of I = I1 × . . . × In
Δ = Δ1 × . . . × Δn, where each Δi is a partition of Ii, a closed interval of R
95. Subinterval J of Δ a partition of I
• The product of one sub-interval from each of the partitions that make up Δ
• Write J ∈ Δ
96. 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
97. 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
98. Upper sum of ƒ for the partition Δ 99. Lower sum of ƒ for the partition Δ 100. Refinement of the partition Δ = Δ1 × . . . × Δn
Π = Π1 × . . . × Πn, where each Πi is a refinement of Δi (meaning Δi ⊆ Πi)
101. Common refinement of the partitions Δ = Δ1 × . . . × Δn and Π = Π1 × . . . × Πn
1 ∪ Π1] × . . . × [Δn ∪ Πn]
102. 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}
103. Integral of an integrable function ƒ 104. Diameter of A
• For X a metric space, and A ⊆ X,
• 105. Mesh of Δ 106. n-cube
A closed interval I = I1 × . . . × In of Rn with the property that l(I1) = . . . = l(In)
107. Cubical partition of Δ
Every subinterval of Δ is an n-cube;
108. Extension by zero of ƒ to B
• For A ⊆ B ⊆ Rn and ƒ : A → R, g : B → R is the extension, defined by
• 109. Characteristic function of the set A ⊆ Rn with A bounded 110. A ⊆ Rn has well-defined volume
• χA is integrable; define
• 111. Translate of A ⊆ Rn by u ∈ Rn
A + u = {a + u : a ∈ A}
112. Jordan domain
A bounded subset D of Rn such that vol ∂D = 0
113. Parallelepiped based at p ∈ Rn spanned by n-vectors u1, . . . um 114. Rotation by θ ∈ R
• Rθ : R2 → R2 the linear function with matrix
• 115. Hyperplane in Rn
• A translation of a n − 1 dimensional sub(vector) space of Rn
• The solution sets of a single linear equation
116. 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
• 117. Uniformly differentiable 118. Jacobian of F : U → Rn differentiable at u ∈ U, with U open in Rn
JF(u) = det DF(u)
 Author: mathprodigy20 ID: 54858 Card Set: Analysis Definitions Updated: 2010-12-14 14:25:21 Tags: Math analysis Folders: Description: List of definitions for Math 342 Show Answers: