Ergänzung zu 9.1 Grundbegriffe In der Vorlesung wurde der

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(4)
(5)
ku + vk2 = (u + v | u + v) = (u | u) + (u | v) + (v | u) + (v | v) = kuk 2 + 2 Re(u | v) + kvk2 .
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ϕ(α) : = ku − αvk2 = (u − αv | u − αv) = (u | u) − α(v | u) − α(u | v) + αα
= kuk2 − (u | v) · (v | u) + [α − (u | v)] · [α − (v | u)]
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= kuk − |(u | v)| + |α − (v | u)| .
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0 ≤ ku − α0 vk2 = ϕ(α0 ) = kuk2 − |(u | v)|2
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Re z ≤ |z|
ku + vk2 ≤ kuk2 + 2|(u | v)| + kv 2 k ≤ kuk2 + 2kuk kvk + kvk2 = (kuk + kvk)2 ,
(4)
(5)
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