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Title: Parallelogram Law
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Series: Hilbert Spaces
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Chapter: Properties of Inner Product Spaces
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YouTube-Title: Hilbert Spaces 4 | Parallelogram Law
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Subtitle on GitHub: hs04_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Definitions in the video: parallelogram law
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $(X, \langle \cdot, \cdot \rangle)$ be an inner product space and $| \cdot |$ the induced norm. What is not correct in general?
A1: $ | x + y |^2 = | x |^2 + | y |^2 $ for all $x,y \in X$.
A2: $ | x | = \sqrt{ \langle x, x \rangle }$ for all $x \in X$.
A3: $| x + y | \leq | x | + | y | $ for all $x,y \in X$.
A4: $| x + y |^2 + | x - y |^2 $ $ = 2 | x |^2 + 2 | y |^2 $ for all $x,y \in X$.
Q2: What is correct?
A1: A Hilbert space is a Banach space where the parallelogram law holds.
A2: In each Banach space, the parallelogram law holds.
A3: There are Hilbert space where the parallelogram law does not hold.
A4: A normed space where the parallelogram law holds is necessary a Hilbert space.
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Last update: 2026-02
