Analysis Definitions

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mathprodigy20
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54858
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Analysis Definitions
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2010-12-14 09:25:21
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Math analysis
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List of definitions for Math 342
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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
  11. Open metric ball about p of radius r in X
    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)

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