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Title: Riesz Representation Theorem
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Series: Functional Analysis
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YouTube-Title: Functional Analysis 15 | Riesz Representation Theorem
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Bright video: https://youtu.be/rKiy6wEiQIk
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Dark video: https://youtu.be/OVvugoEt2ZQ
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Ad-free video: Watch Vimeo video
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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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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: fa15_sub_eng.srt missing
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Timestamps
00:00 Introduction
00:29 Riesz representation theorem
02:45 Proof Existence
07:25 Proof Uniqueness
08:10 Proof Operator Norm
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $(X, \langle \cdot, \cdot \rangle)$ be a Hilbert space and $\ell: X \rightarrow \mathbb{F}$ be a continuous linear functional. What is correct?
A1: There is an element $x_\ell \in X$ with $\langle x_\ell, x \rangle = \ell(x)$ for all $x \in X$.
A2: If $\ell(x) = 0$, then $x$ is orthogonal to every $y \in X$.
A3: $x = 0$ implies $\ell(x) \neq 0$.
A4: For every $x \in X$, we have $\ell(x) = \langle x, x \rangle$.
Q2: Consider the functional $\ell \colon \mathbb{C}^2 \to \mathbb{C}$ defined via $\ell(x) = x_1 + x_2$ for all $x = (x_1, x_2) \in \mathbb{C}^2$. The Hilbert space $\mathbb{C}$ is endowed with the standard inner product. Which statement is not correct?
A1: $\ell(x) = \langle \binom{1}{1}, x \rangle$ for all $x \in \mathbb{C}^2$
A2: $| \ell | = \sqrt{2}$
A3: There exists $x_\ell \neq \binom{1}{1}$ with $\ell(x)= \langle x_\ell, x \rangle$ for all $x \in \mathbb{C}^2$
A4: $\mathrm{ker}(\ell) = \big{\binom{1}{1}\big}^\perp$
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Last update: 2024-10
