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Title: Orthonormal System and Orthonormal Basis
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Series: Hilbert Spaces
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Chapter: ONS and ONB
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YouTube-Title: Hilbert Spaces 10 | Orthonormal System and Orthonormal Basis
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Subtitle on GitHub: hs10_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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Definitions in the video: ONS, ONB, Orthonormal Basis, Orthonormal System, total set
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Timestamps
00:00 Introduction
00:38 Notion: orthonormal
01:51 Definition: (infinite) ONS and ONB
03:39 Notion: total system
06:20 Example: Hilbert space of sequences
09:57 Credits
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $X$ be a vector space with norm $\| \cdot \|$. What does it mean that a set $M \subseteq X$ is dense in $X$?
A1: For all $x \in X$ and all $\varepsilon > 0$, there exists an element $m \in M$ such that $ \| x - m \| < \varepsilon$.
A2: For all $x \in X$ and there is an $\varepsilon > 0$ such that for all $m \in M$ such that $ \| x - m \| < \varepsilon$.
A3: There is an $x \in X$ and there is an $\varepsilon > 0$ such that for all $m \in M$ such that $ \| x - m \| < \varepsilon$.
A4: There is an $x \in X$ such that for all $\varepsilon > 0$ and all $m \in M$ such that $ \| x - m \| < \varepsilon$.
Q2: Let $X$ be a vector space with norm $\| \cdot \|$. What does it mean that a set $M \subseteq X$ is total in $X$?
A1: The subspace $\mathrm{Span}(M)$ is dense in X.
A2: The subspace $\mathrm{Span}(M)$ is equal X.
A3: There is an $x \in X$ and there is an $\varepsilon > 0$ such that for all $m \in M$ such that $ \| x - m \| < \varepsilon$.
A4: For all $x \in X$ and all $\varepsilon > 0$, there exists an element $m \in M$ such that $ \| x - m \| < \varepsilon$.
Q3: Let $X$ be a inner product space. What is the definition of an ONB in it?
A1: It’s an ONS that is also total.
A2: It’s an ONS that is also as basis of the vector space $X$.
A3: It’s an ONS that is also dense in $X$.
A4: It’s an ONS with the property $\langle e_{\alpha} , e_{\beta} \rangle = \delta_{\alpha, \beta}$.
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Date of video: 2025-03-11
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Last update: 2026-10
