• 〈v, v 〉 ≥ 0
نویسنده
چکیده
Problem 1. Hilbert space and the Riesz representation theorem. If you need help with this, it can be found in lots of places – for instance [1] has a nice treatment. i) A pre-Hilbert space is a vector space V (over C) with a 'positive definite sesquilinear inner product' i.e. a function 1 2 and replace v byˆv = v/v, v 1 2 which hasˆv, ˆ v = 1. Thus we may as well assume v, v = 1. Now using the linearity and anti-linearity which follow from the conditions above, This proves Schwarz' inequality. ii) Show that v = v, v 1/2 is a norm and that it satisfies the parallelogram law: (1) v 1 + v 2 2 + v 1 − v 2 2 = 2v 1 2 + 2v 2 2 ∀ v 1 , v 2 ∈ V. Solution. As with the special case of L 2 that I did in class, the triangle inequality follows from Schwarz' inequality: u + v 2 = u + v, u + v ≤ u 2 + 2||u, v| 2 + v 2 ≤ (u + v) 2 .
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تاریخ انتشار 2001