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Title: Approximation Formula
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Series: Hilbert Spaces
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Chapter: Orthogonality
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YouTube-Title: Hilbert Spaces 7 | Approximation Formula
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Subtitle on GitHub: hs07_sub_eng.srt missing
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $X$ be a Hilbert space and $x,y \in X$. What is the correct Pythagorean theorem?
A1: If $x,y$ are orthogonal, then $| x + y |^2 = | x |^2 + | y |^2$.
A2: If $x,y$ are orthogonal, then $| x - y |^2 = | x |^2 - | y |^2$.
A3: If $x,y$ are orthogonal, then $| x + y | = | x | + | y |$.
A4: If $x,y$ are orthogonal, then $| x + y | = | x |^2 + | y |^2$.
Q2: Let $X$ be a normed space, $x \in X$, and $U \subseteq X$ a subset. What is the definition of $\mathrm{dist}(x,U)$, the distance between a point $x$ and a subset $U$?
A1: $\inf_{u \in U} | x - u |$
A2: $\sup_{u \in U} | x - u |$
A3: $\inf_{u \in U} | x + u |$
A4: $\sup_{u \in U} | x + u |$
A5: $\sup_{u \in U} | u |$
Q3: Let $X$ be a Hilbert space and $U \subseteq X$ be a closed subspace. What is an direct result from the approximation formula for $x \in X$?
A1: The infimum $\inf_{u \in U} | x - u |$ is actually a minimum.
A2: The distance between $U$ and $x$ is not defined.
A3: Every $y \in U$ satisfies: $\inf_{u \in U} | x - u | = | x -y |$
A4: $U$ is convex.
A5: $U$ is also a Hilbert space.
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Date of video: 2024-10-18
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Last update: 2026-02
