# Homework Help: Inner product and hermitic product

1. Sep 21, 2012

### damabo

1. The problem statement, all variables and given/known data

prove the triangle inequality ||v+w|| ≤ ||v||+||w||
if (ℂ, V, + , [ , ] ) is a Hermitic space where [ , ] is the Hermitic product.

2. Relevant equations

|[v,w]| ≤ ||v|| . ||w||

$\overline{a+b i}$= a-b i (where i is the imaginary number, a+bi a complex number)

3. The attempt at a solution

||v+w||² = [v+w,v+w]. Since [v+w,v+w] = $\overline{[v+w,v+w]}$, [v+w,v+w] must be a real number. this means that the same rules will apply as in the inner product on the innerproductspace (ℝ, V, +, [ , ] ) ( true ?). so [v+w,v+w] = [v,v] + [w,w] + 2[v,w] ≤ ||v||²+||w||²+2||v|| ||w|| = (||v|| + ||w|| ) ²
this means ||v+w|| ≤ ||v|| + ||w||

Last edited: Sep 21, 2012