2) We used the inner product in a Hilbert space to define its norm. That leaves open the question of whether two different inner products could lead to the same norm. Show that that is not the case, by doing the following. (a) For a real Hilbert space, show that (x, y) ||r + y||? – ||x – y|| (b) For a complex Hilbert space, show that (x, y) = 1 ||a + y||? + i||x + iy||? – ||x – y||? – i||x – iy|?|. -
2) We used the inner product in a Hilbert space to define its norm. That leaves open the question of whether two different inner products could lead to the same norm. Show that that is not the case, by doing the following. (a) For a real Hilbert space, show that (x, y) ||r + y||? – ||x – y|| (b) For a complex Hilbert space, show that (x, y) = 1 ||a + y||? + i||x + iy||? – ||x – y||? – i||x – iy|?|. -
2) We used the inner product in a Hilbert space to define its norm. That leaves open the question of whether two different inner products could lead to the same norm. Show that that is not the case, by doing the following. (a) For a real Hilbert space, show that (x, y) ||r + y||? – ||x – y|| (b) For a complex Hilbert space, show that (x, y) = 1 ||a + y||? + i||x + iy||? – ||x – y||? – i||x – iy|?|. -
Transcribed Image Text:(2) We used the inner product in a Hilbert space to define its norm. That
leaves open the question of whether two different inner products could
lead to the same norm. Show that that is not the case, by doing the
following.
(a) For a real Hilbert space, show that (x, y) = |x + y||² – ||x – y||2|
-
(b) For a complex Hilbert space, show that
(x, y) = 1 ||x + y||? + il|r + iy||? – ||x – y||? – i||x – iy|||.
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Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.
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