Bessel's Inequality
Theorem 1. Let (V,⟨⋅,⋅⟩V,⟨⋅,⋅⟩) be an inner product space, and let S={v1,...,vn}S={v1,...,vn} be an orthonormal subset of VV. Then ∀x∈V∀x∈V, ||x||2≥n∑i=1|⟨x,vi⟩|2.||x||2≥n∑i=1|⟨x,vi⟩|2.
Proof. Let ⟨S⟩=W⟨S⟩=W. Then !∃y∈W,z∈W⊥!∃y∈W,z∈W⊥ such that x=y+zx=y+z by Theorem 1. Thus we have ||x||2=||y||2+||z||2=n∑i=1|⟨y,vi⟩|2+||z||2≥n∑i=1|⟨x,vi⟩|2.◼
Parseval's Identity
Theorem 2. In the notaion of Theorem 1, if V is finite-dimensional and S is an orthonormal basis for V, then ∀x,y∈V, ⟨x,y⟩=n∑i=1⟨x,vi⟩¯⟨y,vi⟩.
Proof. Denote x=∑ni=1⟨x,vi⟩vi,y=∑nj=1⟨y,vj⟩vj. Then we have ⟨x,y⟩=⟨n∑i=1⟨x,vi⟩vi,n∑j=1⟨y,vj⟩vj⟩=n∑i=1⟨x,vi⟩n∑j=1¯⟨y,vj⟩⟨vi,vj⟩=n∑i=1⟨x,vi⟩¯⟨y,vi⟩.◼
Corollary
Corollary. Let ⟨⋅,⋅⟩′ denote the standard inner product of Fn. Then ∀x,y∈V, ⟨x,y⟩=⟨[x]β,[y]β⟩′.