• A vector
space V with an inner product is
called an inner
product space.
•
• Inner product induced norm ||x|| := Ö<x,x>
• Distance between vectors x and y : d(x,y) = ||x - y|| .
• Two vectors x and y are orthogonal if <x,y> = 0, denoted x ^ y.
•
§Properties of Inner
Product:
§
v |<x, y>|£||x|| ||y|| (Cauchy-Schwarz inequality). Equality holds iff x=ay for some constant a or y=0.
v||x+y||2+||x-y||2=2||x||2+2||y||2 (Parallelogram law)
v||x+y||2=||x||2+||y||2 if x ^ y.
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