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Title: Total Orthonormalsystem
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Series: Fourier Transform
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Chapter: Fourier Series
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YouTube-Title: Fourier Transform 9 | Total Orthonormalsystem
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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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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ft09_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: Consider a vector space $V$ with an ONS $(e_i)_{i \in \mathbb{N}}$. What does it mean that the ONS is complete?
A1: For all $x \in V$ we have: $$ \left| x - \sum_{k=1}^n e_k \langle e_k, x \rangle \right| \xrightarrow{ n \rightarrow \infty } 0 $$
A2: For all $x \in V$ we have: $$ \left| x - \sum_{k=1}^n e_k \langle e_k, x \rangle \right| \geq n $$
A3: For all $x \in V$ we have $\langle x, e_k \rangle = 0$ for all $k \in \mathbb{N}$.
A4: For all $k \in \mathbb{N}$ there is a vector $x$ with $\langle x, e_k \rangle = 0$.
Q2: Consider a vector space $V$ with an ONS $(e_i)_{i \in \mathbb{N}}$. What does it mean that the ONS is total?
A1: The linear span of $(e_i)_{i \in \mathbb{N}}$ is dense in $V$.
A2: The linear span of $(e_i)_{i \in \mathbb{N}}$ is equal to $V$.
A3: The linear span of $(e_i)_{i \in \mathbb{N}}$ is of dimension $\infty$.
A4: The linear span of $(e_i)_{i \in \mathbb{N}}$ is of finite dimension.
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Last update: 2025-05
