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Title: Bounded Inverse Theorem and 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 27 | Bounded Inverse Theorem and Example
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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: fa27_sub_eng.srt missing
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Download dark video: Link on Vimeo
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Timestamps
00:00 Introduction
01:30 Counterexample
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $X$ and $Y$ be two Banach spaces and $T: X \rightarrow Y$ be a bounded linear operator. What is not possible at all?
A1: $T$ is invertible and $T^{-1}$ is not continuous.
A2: $T$ is an open map.
A3: $T$ is surjective.
A4: The inverse of $T$ exists and is a bounded linear operator.
A5: $T$ is bijective and an open map.
Q2: Let $X$ and $Y$ be two normed spaces and $T: X \rightarrow Y$ be a bounded linear operator. What is not possible at all?
A1: $T$ is not continuous.
A2: $T$ is an open map.
A3: $T$ is surjective.
A4: The inverse of $T$ exists and is a bounded linear operator.
A5: $T$ is invertible and $T^{-1}$ is not continuous.
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Last update: 2025-09
