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Title: Examples of Hilbert Spaces
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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 2 | Examples of Hilbert Spaces
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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: hs02_sub_eng.srt missing
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Download bright 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: Which of following inner product spaces is not a Hilbert space?
A1: $\mathbb{C}^n$ with standard inner product
A2: $\ell^2(\mathbb{N}, \mathbb{F})$ with inner product $\langle x, y \rangle = \sum_{j=1}^\infty \overline{x_j} y_j$
A3: $C([0,1], \mathbb{F})$ with inner product $\langle f, g \rangle = \int_{0}^1 \overline{f(t)} g(t) , dt$
A4: $\mathbb{R}^n$ with standard inner product
Q2: What is the inner product on the space $L^2(\Omega, \mu)$, which consists of equivalence classes.
A1: $\langle [f], [g] \rangle = \int_{\Omega} \overline{f(\omega)} g(\omega) , d\omega$
A2: $\langle [f], [g] \rangle = \int_{\Omega} f(\omega) g(\omega) , d\omega$
A3: $\langle [g], [f] \rangle = \int_{\Omega} g(\omega) f(\omega) , d\omega$
A4: $\langle [g], [f] \rangle = \int_{\Omega} \overline{f(\omega) g(\omega) } , d\omega$
Q3: What is the inner product on the space $\ell^2(\mathbb{N}, \mathbb{C})$, which consists of functions
A1: $\langle f, g \rangle = \sum_{n=1}^\infty \overline{f(n)} g(n)$
A2: $\langle f, g \rangle = \sum_{n=1}^N \overline{f(n)} g(n)$
A3: $\langle g, f \rangle = \sum_{n=1}^\infty f(n) g(n)$
A4: $\langle g, f \rangle = \sum_{n=1}^\infty f(n)+g(n)$
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Last update: 2026-02
