Analysis 2

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  1. Algebraic Structure
    Let x = ( x, ... , x ), y = ( y, ... , y ) ∈ Rn and α ∈ R.

    1) sum x + y := ( x1 + y1, x2+ y2, ... , xn + yn )

    2) difference x - y := ( x1 - y1, x2 - y2, ... , xn - yn )

    3) product αx := ( αx1, αx2, ... , αx3 )

    4) dot product x · y : = x1y1 + x2y2 + ... + xnyn
  2. Algebraic Definition 1

    i) Euclidean Norm
    ii) L-one-norm
    iii) sup-norm
    iv) distance
    x ∈ Rn.

    • i) Euclidean Norm of x the scalar:
    • Image Upload 1

    • ii) L-one-norm of x is the scalar:
    • Image Upload 2

    • iii) sup-norm of x is the scalar:
    • Image Upload 3

    • iv) distance between two points a, b ∈ Rn is the scalar
    • Image Upload 4
  3. Algebraic Definition

    i) Euclidean Norm : Image Upload 5
    ii) Image Upload 6
    iii) Image Upload 7
    Image Upload 8

    Image Upload 9

    Image Upload 10
  4. Algebraic Definition:

    For a, b,
    i) Orthogonality
    ii) Parallel
    i) a and b are said to be parallel if and only if there is a scalar t ∈ R s.t. a = tb

    ii) a and b are said to be orthogonal if and only if a b = 0.
  5. Algebraic Structure : Inequality

    Image Upload 11

    Image Upload 12
    • Proof:
    • i)
    • Image Upload 13

    • ii)
    • Image Upload 14

    Image Upload 15
  6. Cauchy-Schwartz Inequality

    prove | x ⋅ y | = || x || || y ||
    • Recall:
    • 1) Image Upload 16
    • 2) adding a scalar, t, to get a better estimate to (1) :
    • Image Upload 17

    • Proof: when y = 0, it's trivial.
    • If Image Upload 18, substitute Image Upload 19

    Image Upload 20

    • Image Upload 21
    • Image Upload 22

  7. Topology of Rn: Definition

    i) open ball
    ii) closed ball
    i) For each r > 0, the open ball centered at a of radius r is the set of points

    Image Upload 23

    ii) For each Image Upload 24, the closed ballat centered at a of radius r is the set of points

    Image Upload 25
  8. Topology of Rn: Definition

    i) Open set
    ii) Closed set
    i) A subset V of Rn is said to be open (in Rn) Image Upload 26

    • ii) A subset V of Rn is said to be closed (in Rn)
    • Image Upload 27
  9. Topology: Remark (8.21)

    Prove:

    Image Upload 28
    Let Image Upload 29. Set Image Upload 30. If Image Upload 31, then by the Triangle Inequality and the choice of e,


    Image Upload 32

    Thus, by definition, Image Upload 33. In particular, Image Upload 34
  10. Topology: Remark (8.22)

    Prove:

    Image Upload 35
    Let Image Upload 36 and set Image Upload 37. Then, by definition, Image Upload 38, so Image Upload 39. Therefore, Ec is open.
  11. Topology: Remark (8.22)

    Prove:

    For each n ∈ N, the empty set ∅ and the whole space Rn are both open and closed.
    Since Rn = ∅c and ∅ = (Rn)c, suffices to prove that ∅ and Rn are both open.

    Since the empty set contains no points, "every" point x ∈ ∅ satisfies Be(x) ⊆ ∅ (vacuously). Therefore, ∅ is open.

    On the other hand, since Be(x) ⊆ Rn, ∀ x ∈ Rn and e > 0, Rn is open
  12. Topology : Theorem

    If
    i) {Vα}α∈A any collection of open subsets of Rn
    ii) {Vk : k=1, 2, ..., p} a finite collection of open subsets of Rn
    iii) {Eα}α∈Aany collection of closed subsets of Rn
    iv) {Ek : k = 1, 2, ... , p} a finite collection of closed subsets of Rn,
    v) V is open and E is closed,

    then
    i) Image Upload 40 is open

    • ii)Image Upload 41 is open
    • iii)

  13. Topology : Theorem
Author
ID
138773
Card Set
Analysis 2
Description
Algebraic Structure, Chapter 8 from An Introduction to Analysis by William Wade, 4th edition,
Updated
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