The Intuition behind Hilbert Spaces and Fourier Series

In this video, we generalize Euclidean vector space to obtain Hilbert spaces. In the process, we come across Bessel's inequality and Parseval's identity. The theory of orthonormal sets in Hilbert spaces, leads us to a generalization of the Fourier series of periodic functions.

[TheLethalDomain]

One of the best introductions to Hilbert spaces I have ever seen. This was amazing.

[aditya]

Ive been following many math and science channels over last 6 years but this channel quickly became the best. Thank you sooo much for the great work. I owe u alot!

[arturosedanoquirarte]

It is a great channel, for now it has few subscribers but over time it will grow, do not be discouraged, you are doing a great job.

[SchienexScience]

Wow, I found this treasure channel! Thank u very much❤

[ewthmatth]

I can't believe the highest rated comment here is someone complaining about this benign text-to-speech engine that sounds fine and is perfectly clear.

[jaysn]

Wonderful content, please continue the content! I am so happy to have found your channel at the right time as I am taking a course on functional analysis and I can’t wait for more content.

Also: Maybe a minority opinion here but I prefer the AI voice.

Possible content ideas:
Dual spaces, bidual spaces, injective/surjective embeddings, weak convergence and finally compactness in infinite dimensional vector spaces

[newwaveinfantry]

4:37 - f should be squared and in absolute value under the integral there.

[estebanvasquez-giraldo]

It is indeed intuitive if you have enough background. I am really happy that someone is doing things carefully and for somewhat advanced public. Thank you so much. There is a question I have always had for many years and I would really appreciate it if you could answer: You mention that an orthonormal basis is not a basis. But the u_α are elements of the vector space, and then you say that we can represent any vector as a series, a linear combination of the orthonormal basis elements. How does the orthonormal basis fail to be a basis? Is it because of the uniqueness requirement, can you give a example?

Also related, then you show how the Fourier series is built, and in that case the u_α's are complex exponentials and in that case they are elements of the vector space because we are considering them in a period. But what happens with the Fourier transform. Why can we justify expanding elements of the vector space as linear combination of functions which do not belong to the space, and therefore are not vectors?

[barryzeeberg]

Not sure why the title refers to 'intuition', the math is ok, but in no way intuitive.

[Spix_Weltschmerz-Pucket]

I always tell people, when you find some pedantic physicist, ask them why does the wave equation has to be defined in an infinite Hilbert space? 😅 Great video! Thanks for sharing!

[shrayesraman]

Amazing work! A full video on Legendre polynomials would be amazing!

[munkeypatcher]

Very cool content, gives me warm nostalgia (with some shivers) about my functional analysis course 😅

But it would sound much better without an AI voice. I'd personally prefer the worst mic but the real voice over this AI voice

[Victor-ssbg]

I just discovered your channel and I am pleasantly surprised by the quality of the content! Thank you for the effort put into the creation of the videos, and keep up the good work!

[markwrede]

Deploying Multiplication describes a capacity far smaller than the Infinity available to Addition.

[CrazyShores]

Excellent! What is a great book to read this from? Thanks

[wargreymon]

nice editing, feel like reading a book 🧐

[TrailersReheard]

Channel would be much better without the AI voice

[pietroghsvf]

I'm still waiting for the intuition

[wdobni]

i didn't find any of that to be 'intuitive'