Orthogonal system

$Orthogonal$

a, b

are orthogonal:

⟨ a, b ⟩ = 0

{v_{1}, ... v_{n}}

is an orthogonal system of

V

:

1 \leq i \neq = j \leq n \Rightarrow ⟨ v_{i}, v_{j} ⟩ = 0

{f_{1}, ... f_{n}}

is an orthonormal system of

V

:

1 \leq i, j \leq n \Rightarrow ⟨ f_{i}, f_{j} ⟩ = δ_{ij}

{f_{1}, ... f_{n}}

is an Orthonormal system basis of

V

:

1. {f_{1}, ... f_{n}} is an Orthonormal system of V 2. {f_{1}, ... f_{n}} is an bisis of V

\exists {f_{i} ∣ p + 1 \leq i \leq r, i \in N}, ({f_{1}, \dots, f_{p}, f_{p + 1}, \dots f_{r}}

is an orthonormal basis

))

Orthonormal basis always exits in any finite sub linear space

1. f_{1} = 1∣∣ a_{1} ∣∣ 2. f_{p} = a_{p} - i = 1 \sum p - 1 (a_{p}, f_{i}) f_{i} (1 \leq p \leq r)

Perp of

W

:

W^{⊥} = {x \in V ∣\forall y \in W (⟨ x, y ⟩ = 0)}

(W^{⊥})^{⊥} = W

V = W \oplus W^{⊥}