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Title: Proof of the Approximation Formula
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Series: Hilbert Spaces
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Chapter: Orthogonality
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YouTube-Title: Hilbert Spaces 8 | Proof of the Approximation Formula
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Quiz: Test your knowledge
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Subtitle on GitHub: hs08_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 $U \subseteq X$ be a closed and convex set. What is the correct statement of the approximation formula?
A1: Every $x \in X$ has a unique best approximation $y \in X$ with $| x - y | = \mathrm{dist}(x,U)$.
A2: Every $x \in X$ has a unique best approximation $y \in X$ with $| x + y | = 0$.
A3: Every $x \in X$ lies in $U$.
A4: Every $x \in X$ can be decomposed $y \in X$ and $u \in U$ as $ x = y + u$.
Q2: Let $X$ be a Hilbert space, $x \in X$, and $U \subseteq X$ be a closed and convex set. Recall the proof of the approximation formula. We take a sequence $(u_n)_{n \in \mathbb{N}}$ with the property $u_n \xrightarrow{n \rightarrow \infty} \mathrm{dist}(x,U)$. What can we show?
A1: $(u_n)_{n \in \mathbb{N}}$ is a Cauchy sequence.
A2: $(u_n)_{n \in \mathbb{N}}$ is an unbounded sequence.
A3: $(u_n)_{n \in \mathbb{N}}$ has two accumulation values.
A4: $(u_n)_{n \in \mathbb{N}}$ is always convergent to the zero vector.
Q3: Let $X$ be a Hilbert space, $x \in X$, and $U \subseteq X$ be a closed and convex set. Assume that $\mathrm{dist}(x,U) = 0$. What is a correct conclusion?
A1: $x \in U$
A2: $x = 0$
A3: $U = \emptyset$
A4: $X = {0}$
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Date of video: 2025-01-21
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Last update: 2026-02
