Any inner product induces a norm given by
∥v∥=⟨v,v⟩ Proof. The axioms for norms mostly follow directly from those for inner products.
If u,v∈V and α∈F, then
(i)
∥v∥=⟨v,v⟩≥0, since ⟨v,v⟩≥0 with equality if and only if v=0.
(ii)
∥αv∥=⟨αv,αv⟩=∣α∣2⟨v,v⟩
=∣α∣⟨v,v⟩=∣α∣∥av∥ (iii) The triangle inequality
Cauchy-Schwarz inequality
If V is an inner product space, then
∣⟨u,u⟩∣≤∥u∥∥v∥for all u,v∈V. Equality holds exactly when u and v are linearly dependent.
Using the Cauchy-Schwarz inequality,
∥u+v∥2=⟨u+v,u+v⟩
=⟨u,u⟩+⟨u,v⟩+⟨v,u⟩+⟨v,v⟩
=∥u∥2+⟨u,v⟩+⟨u,v⟩+∥v∥2
=∥u∥2+2Re⟨u,v⟩+∥v∥2
≤∥u∥2+2∣⟨u,v⟩∣+∥v∥2
≤∥u∥2+2∥u∥∥v∥+∥v∥2
=(∥u∥+∥v∥)2 Taking square roots yields
∥u+v∥≤∥u∥+∥v∥, since both sides are nonnegative.
Therefore ∥v∥=⟨v,v⟩ is a Norm.
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