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Algebraic Structure
Let x = ( x, ... , x ), y = ( y, ... , y ) ∈ R^{n} and α ∈ R.
1) sum x + y := ( x_{1} + y_{1}, x_{2}+ y_{2}, ... , x_{n} + y_{n} )
2) difference x  y := ( x_{1}  y_{1}, x_{2}  y_{2}, ... , x_{n}  y_{n} )
3) product αx := ( αx_{1}, αx_{2}, ... , αx_{3} )
4) dot product x · y : = x_{1}y_{1} + x_{2}y_{2} + ... + x_{n}y_{n}

Algebraic Definition 1
i) Euclidean Norm
ii) Lonenorm
iii) supnorm
iv) distance

Algebraic Definition
i) Euclidean Norm :
ii)
iii)

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.

Algebraic Structure : Inequality


Topology of R^{n}: 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
ii) For each _{}, the closed ballat centered at a of radius r is the set of points

Topology of R^{n}: Definition
i) Open set
ii) Closed set
i) A subset V of R ^{n} is said to be open (in R ^{n})
 ii) A subset V of Rn is said to be closed (in Rn)


Topology: Remark (8.21)
Prove:
Let _{}. Set _{.} If _{, } then by the Triangle Inequality and the choice of e,
Thus, by definition, _{}. In particular,

Topology: Remark (8.22)
Prove:
Let and set _{}. Then, by definition, _{}, so _{}. Therefore, E ^{c} is open.

Topology: Remark (8.22)
Prove:
For each n ∈ N, the empty set ∅ and the whole space R^{n} are both open and closed.
Since R^{n} = ∅^{c} and ∅ = (R^{n})^{c}, suffices to prove that ∅ and R^{n} are both open.
Since the empty set contains no points, "every" point x ∈ ∅ satisfies B_{e}(x) ⊆ ∅ (vacuously). Therefore, ∅ is open.
On the other hand, since B_{e}(x) ⊆ R^{n}, ∀ x ∈ R^{n} and e > 0, R^{n} is open

Topology : Theorem
If
i) {V_{α}}_{α∈A} any collection of open subsets of R^{n}
ii) {V_{k} : k=1, 2, ..., p} a finite collection of open subsets of R^{n
}iii) {E_{α}}_{α∈A}any 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) _{ }is open _{
ii) is openiii)
}


