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Title: Isomorphisms?
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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 21 | Isomorphisms
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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: fa21_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
00:50 Example
04:18 Isomorphism
07:31 Examples
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $X,Y$ be two vector spaces. What do we mean be a homomorphism $f: X \rightarrow Y$?
A1: A linear map.
A2: A bijective map.
A3: A injective map.
A4: A surjective map.
A5: A bounded map.
Q2: Let $X,Y$ be two vector spaces. What do we mean be an isomorphism $f: X \rightarrow Y$?
A1: A linear map that is also bijective.
A2: A linear map.
A3: A injective map that is also linear.
A4: A surjective map that is also bounded.
A5: A bounded map.
Q3: Let $X,Y$ be two Banach spaces. What do we mean be an isomorphism $f: X \rightarrow Y$ between Banach spaces?
A1: A linear bijective map with $| f(x) | = | x |$ for all $x \in X$.
A2: A linear map with $| f(x) | = | x |$ for all $x \in X$.
A3: A linear bijective map with $| f^{-1} (y) | = 0$ for all $y \in Y$.
A4: A linear bijective map with $|f \circ f^{-1} (y) | = | x |$ for all $y \in Y$.
Q4: Let $X,Y$ be two Banach spaces with $X = \mathbb{C}^3$ and $Y = \mathbb{C}^4$. Can we have a Banach space isomorphism $f: X \rightarrow Y$?
A1: No, because the dimensions don’t fit.
A2: Yes, one can always find such an isomorphism.
A3: It depends on the norm on $X$ and $Y$.
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Last update: 2025-09
