Introduction to Hilbert Spaces: Important Examples

In this video, I describe two types of Hilbert Spaces, finite-dimensional and infinite-dimensional.

Questions? Let me know in the comments!

[edwardlulofs]

thank you. This verifies my understanding of an aspect of Hilbert spaces that I have searched several books to find. A new physics theory uses finite dimensional instead of the usual infinite. Some consequences of that change are now clear.

[stevewhitt]

Best video on Hilbert Spaces on YouTube. I did have to go study all the other concepts first, but now, with this video I finally understand why QM works best in Hilbert Space. QM is multi-dimensional and requires Complex Vector Spaces. Don't tell Sal, but you deliver the most with best pacing. However, Sal did helped me the most, many years ago with Calculus. Math Basis of QM will stand as a pillar of info. Thanks

[putin_navsegda]

im on my summer holidays and revise math courses ! Thank you for this playlist, it's very clear and understandable !

[wdobni]

its hard to believe that anything so abstruse and arcane and arbitrary as this hilbert space construct actually yields any real or useful data about anything at all, let alone the entire foundation of all matter and energy in the entire universe

[vikrammullachery]

It would be great to see a presentation on Reproducing kernel hilbert spaces

[abcdef]

i can visualize 3 dimension vectors or vector space? why do you need finite but many or infinite dimensional vector spaces? because i cant picture these.

[picobarco]

Hi when someone below said : Bra== " it may be called a conjugate ket". From Linear Algebra there is a term called Dual Space, the Bra-Vector Space is the Dual Space of the Ket-Vector Space. I hope this will clear up the issues surrounding the relationship between Bra and Ket spaces.

[Handbuch]

Would be nice if you used the := notation if something isn't really equal but a definition. It's just a small thing but I think it's a nice tool to really be clear. :)

[abcdef]

why do you need hilbert spaces for quantum mechanics? complex number system is already directional like vectors

[melanieking]

Strictly, don't you also need add in the boundary of L_2{R} for it to be a Hilbert space?

[Tclack]

Is that definition for the inner product of complex vectors correct? (Speaking of finite-dimensional Hilbert space) All the textbooks I've checked have the second vector listed as the one that you find the conjugate of. Unless I'm mistaken and this is something different?

They don't commute if it's complex I believe, so they're not equivalent

[nellvincervantes]

What about the phi function? Is it nessecary to be normalized?

[zhongyuanchen]

I always wonder if there exist any other Hilbert space other than these two extremly intuitive ones.

[huusho]

I think pronouncing it like "brah" is correct as you did

[abcdef]

square integrable function, isnt an integral function just good enough? why does it have to be square?

quad can be integrable too

[pauldorra]

do i need to understand mathematics like these in order to be smart enough to understand human existence on all possible levels?

[haseeblodhi]

Okay, so here is the question.
The inner product of functions like exp^(x) or exp^(2x) exists over a bounded region [a, b] i.e. they are integrable to a finite value within limits. So doesn't that make these function square-integrable and thus part of a Hilbert vector space?

[scw]

Very difficult to speak to people cause of the simple nature of civilian humanity

[georgegrubbs]

Did he say "in-TEG-rable"? Egads. It's "IN-te-grable".

[haseeblodhi]

Okay, so here is the question.
The inner product of functions like exp^(x) or exp^(2x) exists over a bounded region [a, b] i.e. they are integrable to a finite value within limits. So doesn't that make these function square-integrable and thus part of a Hilbert vector space?