-
Title: Hahn–Banach Theorem
-
Series: Functional Analysis
-
Chapter: Core Results in Functional Analysis
-
YouTube-Title: Functional Analysis 25 | Hahn–Banach Theorem
-
Bright video: Watch on YouTube
-
Dark video: Watch on YouTube
-
Ad-free video: Watch Vimeo video
-
Forum: Ask a question in Mattermost
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: fa25_sub_eng.srt missing
-
Download bright video: Link on Vimeo
-
Download dark video: Link on Vimeo
-
Definitions in the video: Hahn-Banach Theorem, separation of points
-
Timestamps
00:00 Introduction
00:20 Hahn-Banach (extension version)
02:03 Applications
-
Subtitle in English (n/a)
-
Quiz Content
Q1: Let $X$ be a Hilbert space and $U \subseteq X$ a closed subspace. Can a continuous linear functional $u^\prime : U \rightarrow \mathbb{F}$ be extended to the whole space as $x^\prime: X \rightarrow \mathbb{F}$ with $| x^\prime | = | u^\prime |$.
A1: Yes, we can just set $x^\prime(x) = 0$ for all $x \in U^\perp$.
A2: No, there are finite-dimensional counterexamples.
A3: No, it does not work in general infinite-dimensional Hilbert spaces.
A4: No, never.
Q2: Let $X$ be a normed space and $U \subseteq X$ be a subspace where a continuous linear functional $u^\prime: U \rightarrow \mathbb{F}$. What is the statement of the Hahn-Banach Theorem?
A1: There is a linear functional $x^\prime: X \rightarrow \mathbb{F}$ with $| x^\prime | = | u^\prime |$ and $x^\prime|_U = u^\prime$.
A2: There is exactly one linear functional $x^\prime: X \rightarrow \mathbb{F}$ with $x^\prime|_U = u^\prime$.
A3: There is exactly one linear functional $x^\prime: X \rightarrow \mathbb{F}$ with $x^\prime|_U = u^\prime$.
A4: There is a continuous linear functional $x^\prime: X \rightarrow \mathbb{F}$ that satisfies $x^\prime(u) = 0$ for all $u \in U$.
A6: There is a $C>0$ such that for all linear functionals $x^\prime : X \rightarrow \mathbb{F} $ we have $ | x^\prime | \leq C$.
Q3: Let $X$ be a normed space and $U \subsetneq X$ be a closed subspace. Can we find a continuous linear functional $x^\prime$ that is non-zero but vanishes on the whole space $U$?
A1: Yes, the Hahn-Banach theorem guarantees it.
A2: No, the Hahn-Banach theorem is only correct in Banach spaces.
A3: No, this only works in Hilbert spaces.
-
Last update: 2025-09
