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Title: Polarization Identity
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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 3 | Polarization Identity
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Bright video: Watch on YouTube
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Dark video: Watch on YouTube
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Ad-free video: Watch Vimeo video
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Original video for YT-Members (bright): Watch on YouTube
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Original video for YT-Members (dark): Watch on YouTube
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: hs03_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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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 a complex inner product space. What can we conclude from the polarization identity?
A1: If we just know $\langle x, x \rangle$ for all $x \in X$, then we know the whole inner product.
A2: If $\langle x, x \rangle \in \mathbb{R}$ for all $x \in X$, then we also have $\langle x, y \rangle \in \mathbb{R}$ for all $x,y \in X$,
A3: If $| x | = 1$, then $\langle x, y \rangle = | y |$.
Q2: Let $(X, \langle \cdot, \cdot \rangle)$ be a real inner product space. What is correct for all vectors $x,y \in X$?
A1: $ 4\langle x, y \rangle $ $= \langle x+y, x+y \rangle $ $- \langle x-y, x-y \rangle $
A2: $ 4\langle x, y \rangle $ $= \langle x+y, x+y \rangle $ $+ \langle x-y, x-y \rangle $
A3: $ 4\langle x, y \rangle $ $ = \langle x-y, x+y \rangle $ $- \langle x-y, x+y \rangle $
A3: $ 4\langle x, y \rangle $ $= \langle x-y, x-y \rangle $ $+ i \langle x+y, x+y \rangle $
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Last update: 2026-02
