1
Functions on ℝ𝑛
The Topology of ℝ𝑛
ℝ𝑛 = {(𝑥1 , 𝑥2 , … , 𝑥𝑛 )| 𝑥𝑖 ∈ ℝ, 𝑖 = 1, … , 𝑛}.
ℝ𝑛 is a vector space with standard basis {𝑒1 , 𝑒2 , … , 𝑒𝑛 } where:
𝑒1 = (1, 0, 0, … ,0)
𝑒2 = (0, 1, 0, … ,0)
⋮
𝑒𝑛 = (0, 0, 0, … ,1).
We define a norm on ℝ𝑛 by:
|𝑥 | = √𝑥12 + 𝑥22 + ⋯ + 𝑥𝑛2 ; 𝑥 = (𝑥1 , 𝑥2 , … , 𝑥𝑛 ).
We can then define a distance on ℝ𝑛 by:
𝑑 (𝑥, 𝑦) = |𝑥 − 𝑦| = √(𝑥1 − 𝑦1 )2 + (𝑥2 − 𝑦2 )2 + ⋯ + (𝑥𝑛 − 𝑦𝑛 )2
where 𝑥 = (𝑥1 , 𝑥2 , … , 𝑥𝑛 ), 𝑦 = (𝑦1 , 𝑦2 , … , 𝑦𝑛 ).
Def. Let {𝑝𝑗 } be a sequence in ℝ𝑛 . We say {𝒑𝒋 } converges to 𝒑 ∈ ℝ𝑛 if for all
𝜖 > 0 there exists a 𝑁 ∈ ℤ+ (ie, the positive integers) such that if 𝑗 ≥ 𝑁 then
|𝑝 − 𝑝𝑗 | < 𝜖.
Def. Let {𝑝𝑗 } be a sequence in ℝ𝑛 . We say {𝒑𝒋 } is a Cauchy sequence if for all
𝜖 > 0 there exists a 𝑁 ∈ ℤ+ such that if 𝑗, 𝑘 ≥ 𝑁 then |𝑝𝑗 − 𝑝𝑘 | < 𝜖.
ℝ𝑛 is complete (i.e. every Cauchy sequence converges) with respect to this
distance function. In addition, ℝ𝑛 is a Banach space (i.e. a complete, normed,
vector space).
, 2
Proposition: Given 𝑥, 𝑦 ∈ ℝ𝑛 , then:
i) |𝑥 + 𝑦| ≤ |𝑥 | + |𝑦| (triangle inequality)
ii) |𝑥 ∙ 𝑦| ≤ |𝑥 | |𝑦| (Cauchy-Schwarz inequality).
Def. A linear transformation, 𝑇: ℝ𝑛 → ℝ𝑚 , is a function such that for all
𝑢, 𝑣 ∈ ℝ𝑛 and 𝑐 ∈ ℝ:
a. 𝑇 (𝑢 + 𝑣 ) = 𝑇(𝑢) + 𝑇(𝑣)
b. 𝑇(𝑐𝑢) = 𝑐𝑇(𝑢).
A linear transformation 𝑇: ℝ𝑛 → ℝ𝑚 can be represented with respect to the
usual basis in ℝ𝑛 and ℝ𝑚 by an 𝑚 × 𝑛 matrix
𝑎11 𝑎12 ⋯ 𝑎1𝑛
𝑎21 𝑎22 ⋯ 𝑎2𝑛
𝑇=( ⋮ )
⋮ ⋮
𝑎𝑚1 𝑎𝑚2 ⋯ 𝑎𝑚𝑛
where 𝑇(𝑒𝑖 ) = ∑𝑚
𝑗=1 𝑎𝑗𝑖 𝑒𝑗 , 𝑒𝑗 = (0, 0, … , 1, 0, 0, … ,0) and the 1 is in the
𝑗𝑡ℎ place.
The coefficients of 𝑇(𝑒𝑖 ) appear in the 𝑖𝑡ℎ column of the matrix.
𝑎11 𝑎12 … 𝑎1𝑛 0 𝑎1𝑖
𝑎21 𝑎22 … 𝑎2𝑛 ⋮ 𝑎2𝑖
𝑇(𝑒𝑖 ) = ( ⋮ ) 1 = ( ⋮ ).
⋮
𝑎𝑚1 … … 𝑎𝑚𝑛 ⋮ 𝑎𝑚𝑖
(0)
Functions on ℝ𝑛
The Topology of ℝ𝑛
ℝ𝑛 = {(𝑥1 , 𝑥2 , … , 𝑥𝑛 )| 𝑥𝑖 ∈ ℝ, 𝑖 = 1, … , 𝑛}.
ℝ𝑛 is a vector space with standard basis {𝑒1 , 𝑒2 , … , 𝑒𝑛 } where:
𝑒1 = (1, 0, 0, … ,0)
𝑒2 = (0, 1, 0, … ,0)
⋮
𝑒𝑛 = (0, 0, 0, … ,1).
We define a norm on ℝ𝑛 by:
|𝑥 | = √𝑥12 + 𝑥22 + ⋯ + 𝑥𝑛2 ; 𝑥 = (𝑥1 , 𝑥2 , … , 𝑥𝑛 ).
We can then define a distance on ℝ𝑛 by:
𝑑 (𝑥, 𝑦) = |𝑥 − 𝑦| = √(𝑥1 − 𝑦1 )2 + (𝑥2 − 𝑦2 )2 + ⋯ + (𝑥𝑛 − 𝑦𝑛 )2
where 𝑥 = (𝑥1 , 𝑥2 , … , 𝑥𝑛 ), 𝑦 = (𝑦1 , 𝑦2 , … , 𝑦𝑛 ).
Def. Let {𝑝𝑗 } be a sequence in ℝ𝑛 . We say {𝒑𝒋 } converges to 𝒑 ∈ ℝ𝑛 if for all
𝜖 > 0 there exists a 𝑁 ∈ ℤ+ (ie, the positive integers) such that if 𝑗 ≥ 𝑁 then
|𝑝 − 𝑝𝑗 | < 𝜖.
Def. Let {𝑝𝑗 } be a sequence in ℝ𝑛 . We say {𝒑𝒋 } is a Cauchy sequence if for all
𝜖 > 0 there exists a 𝑁 ∈ ℤ+ such that if 𝑗, 𝑘 ≥ 𝑁 then |𝑝𝑗 − 𝑝𝑘 | < 𝜖.
ℝ𝑛 is complete (i.e. every Cauchy sequence converges) with respect to this
distance function. In addition, ℝ𝑛 is a Banach space (i.e. a complete, normed,
vector space).
, 2
Proposition: Given 𝑥, 𝑦 ∈ ℝ𝑛 , then:
i) |𝑥 + 𝑦| ≤ |𝑥 | + |𝑦| (triangle inequality)
ii) |𝑥 ∙ 𝑦| ≤ |𝑥 | |𝑦| (Cauchy-Schwarz inequality).
Def. A linear transformation, 𝑇: ℝ𝑛 → ℝ𝑚 , is a function such that for all
𝑢, 𝑣 ∈ ℝ𝑛 and 𝑐 ∈ ℝ:
a. 𝑇 (𝑢 + 𝑣 ) = 𝑇(𝑢) + 𝑇(𝑣)
b. 𝑇(𝑐𝑢) = 𝑐𝑇(𝑢).
A linear transformation 𝑇: ℝ𝑛 → ℝ𝑚 can be represented with respect to the
usual basis in ℝ𝑛 and ℝ𝑚 by an 𝑚 × 𝑛 matrix
𝑎11 𝑎12 ⋯ 𝑎1𝑛
𝑎21 𝑎22 ⋯ 𝑎2𝑛
𝑇=( ⋮ )
⋮ ⋮
𝑎𝑚1 𝑎𝑚2 ⋯ 𝑎𝑚𝑛
where 𝑇(𝑒𝑖 ) = ∑𝑚
𝑗=1 𝑎𝑗𝑖 𝑒𝑗 , 𝑒𝑗 = (0, 0, … , 1, 0, 0, … ,0) and the 1 is in the
𝑗𝑡ℎ place.
The coefficients of 𝑇(𝑒𝑖 ) appear in the 𝑖𝑡ℎ column of the matrix.
𝑎11 𝑎12 … 𝑎1𝑛 0 𝑎1𝑖
𝑎21 𝑎22 … 𝑎2𝑛 ⋮ 𝑎2𝑖
𝑇(𝑒𝑖 ) = ( ⋮ ) 1 = ( ⋮ ).
⋮
𝑎𝑚1 … … 𝑎𝑚𝑛 ⋮ 𝑎𝑚𝑖
(0)
