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Title: Proof of Jordan-von Neumann Theorem
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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 5 | Proof of Jordan-von Neumann Theorem
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Subtitle on GitHub: hs05_sub_eng.srt missing
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Subtitle in English (n/a)
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Quiz Content
Q1: What is the statement of the Jordan-von-Neumann theorem?
A1: A normed space where the parallelogram law holds is necessary an inner product space.
A2: Each normed space is also an inner product space.
A3: There are norms which are not induced by inner products.
A4: Each norm is induced by an inner product.
A5: A norm that satisfies the parallelogram law can have two different inner products that induce it.
Q2: Which property of the inner product needs the parallelogram law in the proof of the Jordan-von-Neumann theorem?
A1: Linearity in the second argument.
A2: The positive definite property.
A3: Symmetry in the real case.
A4: Conjugate symmetry in the complex case.
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Last update: 2026-02
