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

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

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

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

    i) Euclidean Norm : Image Upload
    ii) Image Upload
    iii) Image Upload
    Image Upload

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  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

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    • Proof:
    • i)
    • Image Upload

    • ii)
    • Image Upload

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  6. Cauchy-Schwartz Inequality

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

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

    Image Upload

    • Image Upload
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  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

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

    Image Upload
  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

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

    Prove:

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


    Image Upload

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

    Prove:

    Image Upload
    Let Image Upload and set Image Upload. Then, by definition, Image Upload, so Image Upload. 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 is open

    • ii)Image Upload is open
    • iii)

  13. Topology : Theorem
Author:
shm224
ID:
138773
Card Set:
Analysis 2
Updated:
2012-03-01 16:43:06
Tags:
real analysis algebraic Structure
Folders:

Description:
Algebraic Structure, Chapter 8 from An Introduction to Analysis by William Wade, 4th edition,
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