Orthogonal/Orthogonality (·⟂·)

• given an inner product space (𝑉,𝐹,⟨·,·⟩):
  • vectors 𝑢∊𝑉 and 𝑣∊𝑉 are orthogonal if the inner product ⟨𝑢,𝑣⟩ = 0 (denoted as 𝑢⟂𝑣)
• given a vector space (𝑉,𝐹) with bilinear form (𝐵):
  • vectors 𝑢∊𝑉 and 𝑣∊𝑉 are orthogonal if 𝐵(𝑢,𝑣) = 0
• subsets 𝐴,𝐵⊆𝑉 are orthogonal if 𝑎⟂𝑏 for all 𝑎∊𝐴 and 𝑏∊𝐵 (denoted as 𝐴⟂𝐵)

Orthogonal Sets

• a set of vectors form an orthogonal set if all vectors in the set are mutually orthogonal
• an orthogonal set which forms a basis is called an orthogonal basis
• can be represented as an orthogonal matrix

Orthogonal - Examples

Orthogonal - In Relation to Norm & Pythagorean Theorem

If 𝑎⟂𝑏, then (||𝑎+𝑏||_⟨·,·⟩)² = (||𝑎||_⟨·,·⟩)² + (||𝑏||_⟨·,·⟩)²

Where ||·||_⟨·,·⟩ is the Inner Product Norm

Orthogonal Complement  (·^⟂)

• for 𝐴⊆𝑉, the orthogonal complement of 𝐴 is 𝐴^⟂ = { 𝑥 |  ⟨𝑥,𝑎⟩ = 0 for all 𝑎∊𝐴, 𝑥∊𝑋}
  • 𝐴^⟂ is always a subspace in 𝑋
  • 𝐴^⟂ is always closed in 𝑋

Orthogonal Complement - 𝐴^⟂ is always a subspace in 𝑋

TODO

Orthogonal Complement - 𝐴^⟂ is always closed in 𝑋

Given:

• an inner product space (𝑉,𝐹,⟨·,·⟩)
• 𝐴⊆𝑋

Then:

• 𝐴^⟂ is closed

Proof:

• Let (𝑥_𝑛)_𝑛∊ℕ ⊆ 𝐴^⟂ be a sequence with limit 𝑥̅∊𝑋 
• ⟨𝑥_𝑛,𝑎⟩ = 0 for all 𝑎∊𝐴
• 𝑙𝑖𝑚_𝑛→∞⟨𝑥_𝑛,𝑎⟩ = 0 for all 𝑎∊𝐴
• ⟨𝑥̅,𝑎⟩ = 0 for all 𝑎∊𝐴
• 𝑥̅∊𝐴^⟂
• Since all sequences converge to elements 𝑥̅ that exist in 𝐴^⟂
• then by definition, 𝐴^⟂ is closed