-
Title: Introductions and Cauchy-Schwarz Inequality
-
Series: Hilbert Spaces
-
Chapter: Properties of Inner Product Spaces
-
YouTube-Title: Hilbert Spaces 1 | Introductions and Cauchy-Schwarz Inequality
-
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: hs01_sub_eng.srt missing
-
Download bright video: Link on Vimeo
-
Download dark video: Link on Vimeo
-
Timestamps
00:00 Introduction
00:45 Network for the video courses
01:40 Prerequisite for the course
02:27 Topics in Hilbert Spaces
03:53 Definition for inner product spaces
06:53 Pre-Hilbert space as an alternative name
07:11 Cauchy-Schwarz inequality
07:53 Proof of Cauchy-Schwarz
11:20 Norm on inner product spaces
11:57 Definition of Hilbert space
12:34 Credits
-
Subtitle in English (n/a)
-
Quiz Content
Q1: Let $\langle \cdot, \cdot \rangle$ be an inner product on $\mathbb{C}^n$. What is not correct in general?
A1: $\langle x, y \rangle = 0$ implies $x = y$
A2: $\langle x, x \rangle = 0$ implies $x = 0$
A3: $\langle x, 0 \rangle = 0$ for all $x \in \mathbb{C}^n$
A4: $\langle x, i y\rangle = - \langle i x, y\rangle$ for all $x,y \in \mathbb{C}^n$
Q2: Let $(X, \langle \cdot, \cdot \rangle)$ be an inner product space and $x,y \in X$ with $\langle x, x \rangle = \langle y, y \rangle = 1$. What is always correct?
A1: $|\langle x, y \rangle| \leq 1$
A2: $|\langle x, y \rangle| \geq 1$
A3: $|\langle x, y \rangle| = 1$
A4: $|\langle x, y \rangle|^2 \geq \langle x, x \rangle + \langle y, y \rangle $
A5: $|\langle x, y \rangle|^2 \geq \langle x, x \rangle \cdot \langle y, y \rangle $
A6: $|\langle x, y \rangle|^2 \leq \frac{1}{2} \langle x, x \rangle \cdot \langle y, y \rangle $
Q3: Let $(X, \langle \cdot, \cdot \rangle)$ be an inner product space with $\dim(X) \geq 1$. Is the map $x \mapsto \langle x, x\rangle$ a norm?
A1: Yes it is!
A2: No, it’s not positive definite.
A3: No, it’s not homogenous.
A4: One needs more information.
-
Last update: 2026-02
