Past exam of the analysis course of the University of Oxford / 2007 / B / 5 / b / i

Solution

Put

Q (u) = ∥ u ∥^{2}

and

B (u, w) = \frac{1}{4} [Q (u + w) - Q (u - w)]

. Then

B (0, w) = 0

u + w, v

and to

u - w, v

and subtracted, gives

B (u + v, w) + B (u - v, w) = 2 B (u, w) .

Taking

v = u

shows

B (2 u, w) = 2 B (u, w)

. Replacing

u, v

in the identity by

(u + v) /2, (u - v) /2

now gives

B (u, w) + B (v, w) = 2 B ((u + v) /2, w) = B (u + v, w) .

The proposed complex pairing is

P (u, w) = B (u, w) + i B (u, i w)

. Applying these same identities with

w

and

i w

proves both asserted midpoint and additivity equalities for $P$ .

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