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Title: Dual Space - Example
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Series: Functional Analysis
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Chapter: Core Results in Functional Analysis
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YouTube-Title: Functional Analysis 23 | Dual Space - Example
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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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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: fa23_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
00:00 Introduction
00:31 Example
03:13 Proof
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $X$ be the normed space $\ell^2(\mathbb{N})$. Is there a linear isometric bijection $X \rightarrow X^\prime$?
A1: Yes!
A2: No, there is only a conjugate-linear isometric bijection.
A3: No, the dual space is completely different.
A4: One needs more information.
Q2: Let $X$ be the normed space $\ell^3(\mathbb{N})$. Which of the following spaces is isometrically isomorphic to $X^\prime$?
A1: $ \ell^q(\mathbb{N}) $ with $q = \frac{3}{2} $
A2: $ \ell^q(\mathbb{N}) $ with $q = \frac{2}{3} $
A3: $ \ell^q(\mathbb{N}) $ with $q = \frac{1}{3} $
A4: $ \ell^q(\mathbb{N}) $ with $q = 3 $
A5: $ \ell^q(\mathbb{N}) $ with $q$ satisfying $ q + 3 = q^{-1} $
Q3: Let $X = \ell^p(\mathbb{N})$ and $Y = \ell^{q}(\mathbb{N})$ where $p^{-1} + q^{-1} = 1$. Which map gives us an isometric ismorphism between $Y$ and $X^\prime$?
A1: $y \mapsto \langle \overline{y}, \cdot \rangle_{\ell^2}$
A2: $y \mapsto \langle \cdot, y \rangle_{\ell^2}$
A3: $y \mapsto \sum_{j=1}^\infty y_j$
A4: $y \mapsto \Big(x \mapsto \sum_{j=1}^\infty \overline{y_j} x_j \Big)$
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Last update: 2025-09
