An Introduction to Hilbert Spaces

In this video, I introduce the Hilbert Space and describe its properties.
Questions? Let me know in the comments!
Prereqs: Previous video on vector spaces, knowledge of what real and rational numbers are.
Lecture Notes: drive.google.c...
Patreon Link: www.patreon.co...

[@whitehorse1959]

Thanks for trying. I still have no idea.

[@FacultyofKhan]

Maybe the introductory video might help? This one isn''t the first on my playlist: kzbin.info/www/bejne/nIfJoWtoe9WGhZo

[@31337flamer]

A Hilbert Space is a standardized space. Mathematicians are crazy and always try to explore their new discveries.. they created crazy spaces with gaps and rounded twisted spaces .. anything u can imagine like a donut etc... to have a common base, it was necessary to define a totally normal space wich has no gaps, is smooth and straight in every direction. so they came up with 4 statements about distances and multiplication that have to be true if u want to have a n-Dimensional totally normal space without any "bumps gaps and bendings" .. and thats called a Hilbert Space.. if u vary the 4 statements or delete one of them or add more u get a different Space ( from: vectorspace -> normed vectorspace -> banachspace -> hilbertspace -> to: euclidean space) every step standardizes the space a little more. the euclidean space that u know from school math is a hilbertspace with finite dimensions... ;) its just a special case of it

[@StuartBarter]

I feel the same hahaha

[@tokuchi7726]

@@31337flamer thanks

[@Peter_1986]

I haven't watched these videos yet, but based on what my rather lame math book is trying to say it seems as if a Hilbert Space is basically just a vector space where functions are defined as vectors, and where functions are considered "orthogonal" if they have an inner product of zero. It's something like that.

[@VishalGupta-lw9dk]

Wow.... You made sense in a really elegant, simple to understand and standard way..,.. Pls make more such videos on Quantum mechanics

[@LydellAaron]

Thank you. Very helpful in adding to understanding. Hilbert spaces are a countably dense vector space which has an inner product that can reduce the vectors to a scalar, obeys conjugate symmetry rules, and is positive definite.

[@malikialgeriankabyleswag4200]

Also the metric because a Vector space doesnt necessarily need a metric, so it also has that distance measure between the vectors

[@christheother9088]

This is helpful. Frustrating to listen to some physics talks where I start to get confused about three words in. Apparently mathematical spaces are not just abstractions anymore, they ARE reality.

[@mastershooter64]

well more accurately they're descriptions of reality

[@derek8482]

Thank you for this video, clears some things for me. I cannot read all the stuff in the book, i just can't go through it without getting distracted and bored.

[@Asdun77]

God bless you for this great explanation

[@azimsofi7360]

wow, this is simpler than a normal textbook, yet complete. thank you!

[@FacultyofKhan]

Glad you like it!

[@tremainerobertson2794]

Yes, it is complete, i.e., no gaps in his explanation! He is indeed keeping it real (pun intended...😂)

[@info-hub457]

Amazing lecture sir only one of its kind. If have searched so much but all explanations were very difficult to understand. You are amazing

[@DialecticRed]

:/ came here because I know hilbert spaces are important for understanding spinors which are an important part of quantum field theory. I don't have the technical knowledge to understand this one right now though - I'm just looking to get a general understanding of QFT and General Relativity so I can really understand the fundamental rift between them, but right now I'm through my first semester of AP Physics II so a lot of this is going above my head, although I did notice a couple of parallels with the mathematical concept of quaternions. I probably didn't get a lot out of this video b/c I'm viewing it somewhat out of context, and I'm sure what your doing is great because it seems like its helping lots of people, but unfortunately I'm just not yet at the point where I can truly understand and appreciate Hilbert Space.

[@RazorM97]

in the context of physic a hilbert space is a space of all possible quantum functions (represented as vectors, which actually have mathematical properties, like the ones mentioned in the video) the completeness property allows us to describe the time evolution of a quantum system using the Schrödinger equation, while the unitarity property ensures that the total probability of all possible measurement outcomes is conserved. An example of the time evolution of a quantum system using the Schrödinger equation and the unitarity property is the decay of a particle. For example, consider the decay of an unstable particle such as a neutron. The neutron is initially in a state described by a wave function, which is a complex-valued function in the Hilbert space of possible quantum states. As time goes on, the wave function evolves according to the Schrödinger equation, which gives the probability amplitude for the particle to be in any given state at any given time. Completeness: The space of all square-integrable functions on the interval [0, 1] is a complete Hilbert space with the inner product defined as: ⟨f, g⟩ = ∫₀¹ f(x) g(x) dx Norm: The space of all square-integrable functions on the interval [-π, π] is a Hilbert space with the norm defined as: ||f|| = (∫₋πᴨ |f(x)|² dx)¹/² Orthogonality: In the space of all square-integrable functions on the interval [0, 1], the functions sin(nπx) and sin(mπx) are orthogonal for different values of n and m. Unitarity: The Fourier transform on the space of all square-integrable functions on the interval [-π, π] is a unitary operator. also anyone reading , this might also help kzbin.info/www/bejne/lZythoiun6-SfLs

[@thedownzguy]

This dude knows what's up.

[@leonardoribeirodesouza3727]

That's awesome. I like so much when you work in another spaces.

[@rogerganga]

Am I the only one who came here after Siraj Raval's "complicated Hilbert space"? :/

[@androukos6441]

No you are not ;)

[@Wintermute--]

lol that "complicated Hilbert space" had me dying.

[@FacultyofKhan]

Aww damn it, I was so busy writing non-plagiarized research papers that I missed the meme train! Oh well.

[@maxcrous]

Legendary :')

[@MikaTarkela]

I came here from Lex Fridman's podcast with Sean Carroll :)

[@adamgm84]

That feel when you enter a video with 1 question but leave with 46.

[@gillesbaumann675]

really good video! Good structure and clearly explained!

[@FacultyofKhan]

Thank you! Glad you liked it!

[@crazyboii3236]

Nice Work! I am very grateful for this. I love your channel. Keep it up.

[@keppela1]

This video should really be titled "An introduction to Hilbert spaces for grad students".

[@chasethescientistsaturre5009]

I think a better name for this series the mathematics of quantum mechanics

[@forhadcsekuet]

Examples of properties (b) and (c) are reversed. It should be like this, (b)Linearity : = a + b and (c) Antilinearity: = a* + b* Please make me correct if I am wrong. Ref: en.wikipedia.org/wiki/Hilbert_space

[@danielescotece7144]

I remember it as a banach space with inner product

[@shaunhumphreys6714]

hilbert space features as a n dimensional space in many worlds interpretation of quantum mechanics-with the parallel world splitting of reality-decoherence process that follows the Schrodinger equation. It feels like this space is a space within a space i.e. still in our spacetime but this is like a confined space, albeit since the size between elementary particles and an atom can be more vast than distances in our universe in terms of difference in size. so if one were hypothetically able to shrink down to a quark size, i would imagine being in a n dimensional void of some kind, except one with loads of virtual particles appearing and disappearing all the time. and loads of particles around including lots of photons. if i went into a biological living thing i would see lots of fermions of course. this space is normally described as an abstract mathematical space. and then like the observable universe bubble to the universe beyond that, there is the gap between fermions and bosons, and planck length. But for me mathematics is real, just as geometry is real, So I'm interested in the reality of such a hilbert space. I'm imagining that in that quantum world they are as real as our classical space seems to us,even though our reality is a very persistent illusion in many ways.

[@fbulderin2415]

Please link to the previous video in the notes. I got here searching for fock space and the video starts by referring to vector space, which I know nothing about, so a link to they previous video would help.

[@curtpiazza1688]

Thanx!

[@nikilragav]

Is this your actual handwriting? It's beautiful haha. What program do you use to write this in?

[@FacultyofKhan]

Yup! I use smoothdraw!

[@saurabhdhabeknight]

14nov 2019 Rip sir Vashisht Narayan Singh 🙏😔😭

[@richardshane456]

Pi is interesting, as from a lay person observing the language describing rational numbers as what I perceive as a displacement theorem that may describe the behavior of rational numbers simply as a displacement theorem of numbers ... Back to pi, which has been described as a irrational number maybe a miss observation of function as a irrational number. What pi, as I observe, using the displacement theorem of mathematics, I ask what natural function would create a irrational number, after pondering for sometime, my observation is pi is not describing an irrational number as explained by any mathematics as a label, I propose what function we are observing is a spiral, and if u think deep enough our universe is well defined as a kinetic energy of spin... I suggest the spin is the natural electro motive function of electrons and all particles interacting against the resistance we call entropy... I also might suggest that all well defined spiral galaxies emits leading and trailing singularities that we have observed as the culmination of the theorem called the big bang from singularities Any comments or thoughts, if you are a mathematician I'll show you a simple observation of your mathematics by your mathematics how mathematics will never be able to describe reality nor the universe Right now it seem it is mental masturbation for chemically induced euphoria for the participants even though you may have created rudimentary products that mimic our perceived reality does not prove your mathematics has any value other than counting with metrics in a technology inquisitive perceived reality created by those who have been able to harness an economic system created based on simple numbers of counting and interest.... Like I said any mathematician who would like to see what I have discovered about mathematics and the notion mathematics has to be the only precise way to live....lol

[@joshuakyanaalampour]

Amazing resource you are! Thank you so very much!

[@abcdef2069]

so hilbert space is complete and has no gaps, can i say hilbert space is real number space? since as you said real number has no gaps and complete.

[@sanjursan]

Well, not quite. Being like the reals is necessary but not sufficient. Like all Ohioans are Americans, but not all Americans are Ohioans. In this case all real numbers are also complex numbers, but not all complex numbers are real numbers. There is more, but I hope this helps a bit.

[@erickgomez7775]

Does the cauchy sequence needs to converge inside the gilbert space? I believe we just need the cauchy sequence to have elements arbitrarily close and inside the gilbert space. No need to converge.

[@chessmanwriter12]

Nice explanations!

[@EW-mb1ih]

nice video but I guess I lack some context about Hilbert space to really understand the importance of these rules

[@kartikvarshney9257]

Always awesome

[@ujjwal_005]

is there a reason as to why hilbert spaces need to be complete and separable?

[@tacopacopotato6619]

I'm a little thrown for a loop here. If rationals are incomplete because reals fill in the gaps, shouldn't we consider reals incomplete because its gaps are filled by Conway's surreal numbers? I suppose there's probably a rigorous meaning that is somewhat lost in order to appeal to intuition. Thanks btw

[@goketesh]

El mejor! gracias!!!

[@richardd7614]

I kind of feel that the only reason i understand the vid is because i already am studying quantum mechanics and am familiar with the mathematics involved in solving a wavefunction, not sure who the audience is for this.

[@chetanreddyn4846]

nice video it was really helpful

[@fcalin21]

My head exploded .

[@qqw1-101]

I get all the other stuff, I just don't get how scalars can have complex conjugates even though the complex conjugate is only for complex numbers...

[@vetiarvind]

Cool..but what is an inner product? (some function of 2 vectors that make a scalar) but what is a complex conjugate? oh man i'm so lost.

[@elinashishkina8610]

The formula e) is written uncorrectly.

[@HDQuote]

how can we have positive definiteness (d) if the inner product returns a complex number? We cannot use < and > in C after all.

[@sherllymentalism4756]

Quote you can, take the magnitude x

[@JDechnics]

no wonder Siraj call this stuff complicated...

[@semimathi6018]

this is so good thanks!

[@tejomaya4817]

Thanks a lot for this video

[@Tyelnd]

I didn't find any video in Spanish that explains me as well as this one.

[@abcdef2069]

psi 1 psi 2.... these are vector 1 and vector 2 as you said. cauchy sequence , n goes infinity, psi n or psi-infinity , what does it mean? why do you need infinite vectors

[@alannolan3514]

brilliant

[@beng2617]

0:21 "One of these additional properties is that a Hilbert space, in addition to the vector addition and the scalar multiplication operation, also has an inner product operation." Well guys, I think that's enough physics for tonight.

[@integralogic]

You think that's bad I came here after listening to a song called "adrift in hilbert space " expecting an explanation and now i have a headache

[@pu3nuraizzatkamal170]

if it involves matrices. how complex conjugate will be? will it be the transpose matrix? how we want to prove sequence is converge in Hilbert space?

[@FacultyofKhan]

A complex conjugate of a matrix is typically a conjugate transpose (aka. Hermitian conjugate), so you take the conjugate of the matrix (i.e. conjugate each element), then take the transpose (or vice-versa).

[@nowornever5598]

Thanks

[@SuperDeadparrot]

Can a Hilbert space be discrete?

[@primozkocevar7722]

Great video! P.S. I think you meant to say positive semi-definite there as 0 is actually possible.

[@kenkal1829]

hilbert is what happens when the human mind has no answer for reality! Until we understand the concept of infinity we can never know, period!!!

[@francescobeccarini1218]

It’s positive because is 0 only when you take the vector zero and the vector zero doesn’t count for the definition of positive

[@abcdef2069]

why do you need complex conjugate symmetry? is it because complex numbers? then why do you need complex numbers? normal vectors always commute vec A dot vec B = vec B dot vec A

[@AbhishekAnandkullu]

abc def Because if they are not complex conjugate, the self inner product doesn't have to be a real number. In abstract formulation, say (A,B) defines the inner product of A and B if (A,B) = C1 (some complex number) also (A,B) = (B,A) = C1 then (A,A) = C2 (another complex number) Now, we want the norm of a physical quantity (a vector from H space) to be a positive number. It can't be a complex number. To get this condition satisfied, we add the conjugate symmetry.

[@haya4895]

thanx alot!

[@thdgus7895]

Il y a plusieurs des questions que concernant la série de reiman F(X)=1+1/2^x +1/3^x++---++++---+-vers l'infini F(X)=1/1-(e^x)^-ln2(1+(e^x)^-c) avec c~0•29 cet résultats montrent que quelque soit X appartient à R la série est converge de même la série harmonie de même dans l intervalle [0'1] la série est converge puisque les séries d exponentielle n ont pas soumis l ordre des ensembles pour la loi d addition 'plusieurs des questions proposer sur ce contexte incompréhensible

[@led98zg]

Thank you. But why do we have the necessity to introduce the Hilbert space? Do we have some mathematical limits in terms of computing in a infinite-dimensional space?

[@devd_rx]

we don't need to introduce infinite dimensions I guess, if we just make each point on the dense space a dimension in itself and store values like that, there won't be difference in the concept, i just feel like thats how it should work, don't take my word for it

[@mohamedyahia4422]

hi i want know the reason why prob b is linear and c is not linear? and thank you

[@pouncingferret7767]

What does he mean by complex conjugate of a scalar? Aren't Complex numbers supposed to be vectors while Scalars are just magnitudes?

[@FacultyofKhan]

Complex numbers can be treated as 2-dimensional numbers (depending on the context), but in this context, I've used them as scalars. So the complex conjugate of z = a + bi is z* = a - bi, where i^2 = -1.

[@tamasemri1261]

Properties b) and c) are the other way around; the inner product is linear w.r.t. its first and antilinear w.r.t. its second argument. Otherwise the video is great!

[@Nellak2011]

Wouldn't the space be complete just by the fact that it's separable, like the real numbers? I don't understand why that is listed as 2 separate properties when they are basically synonomous.

[@FacultyofKhan]

Not quite. Completeness and separability are two different ideas: for instance, the set of irrational numbers is separable (consider m*sqrt(2)/n where m and n are integers as a countably dense subset to prove this). However, the set of irrational numbers is clearly incomplete (there are obvious gaps between groups of irrationals). Hope that helps!

[@seanki98]

Can't we say that R is incomplete as it doesn't include imaginary numbers too? Or does it not matter as in a sense they are orthogonal?

[@FacultyofKhan]

The informal way of referring to completeness means that the set has no 'gaps'. Real numbers exist on a number line, and since they include the entire number line, they don't have gaps. Imaginary numbers aren't anywhere on that number line so they don't count. Instead, imaginary numbers are 'outside' that real number line and exist on a complex plane. Thus, when referring to the completeness of reals, we exclude imaginary numbers, because they don't create any gaps on the real number line. Hope that helps!

[@neildhan]

You'd also fall hard, in trying to find a Cauchy sequence of reals that converges to a complex number with non-zero imaginary part. So regardless of how you picture it, in terms of gaps, the actual definition of completeness is what really matters.

[@edwardlulofs444]

Sean Thrasher . Yes. R is a subset of C.

[@pavanmeena8536]

well, R is the set of all real number that's why imaginary numbers are not included in this set.so no we can't say that it is incomplete .

[@davidwright8432]

The word 'incompete' is ambiguous between technical mathusage, and everyday speech. R is 'complete' in the technical math sense, but of course you can extend the notion of real number ot the complex case, that to the quaternion case, and that to the octonian! Mercifully that's as far as it goes And you can ask all the questions of the complex and up numbers, that you can of the boringly 1-dimensional reals. To put your mind at rest, there are no systems with three, five or seven complex bases!

[@yadavakrishnanp]

In the books, I see the scalar/inner product between elements of the Hilbert space needs to be strictly positive, but then they say between different elements, it can be complex(doesnt this be it can be negative number too with 0i), and within itself(i.e inner product with the element itself) it should be positive or 0. These are bit ambiguous. Can you plz clarify these. I am ok with the self on self inner product..but not comfortable with the inner product with different elements.

[@yadavakrishnanp]

and moreover cant the complex number be negative too like -1-i ? Can you plz clarify with some example..thank you.

[@FacultyofKhan]

The inner product with different elements in a Hilbert Space doesn't have to be positive. For example, R^2 (the space of 2-tuples of real numbers) is a Hilbert Space. 2 possible vectors in this Hilbert space are [-1 1] and [1 -1]. If you do an inner product between them, the answer is -2, which is clearly negative. Within elements, however, the inner product of a vector with itself is either positive or zero. I go over examples of Hilbert Spaces in this video, maybe you might find it useful as well: kzbin.info/www/bejne/q5KQimehZcykbtk

[@yadavakrishnanp]

Faculty of Khan alright. got it. thnx. I was just bit unsure as that wasn't clearly stated in the book. will probably be asking more questions in the days to come. thnx for ur videos etc

[@asmaiqbal402]

Can you please make a video on complex linear spaces??

[@jaywulf]

If the emergent space time is based on Hilbert space functions and Hilbert spaces are complete, does that then mean that the universe is not infinite? Or is it just countable infinite?

[@Cosmalano]

Why would it?

[@musicalwanderings7380]

Separable...from what????

[@moses2061]

Could you share the reference book you are using?

[@Graham_Wideman]

2:15 and on: "R is separable because Q is a countable, dense subset". Please explain the "because". What is the meaning of "separable"? What is it that can be separated? And separated into what separate parts? As far as this video goes, "separable" might just as well be called "bliggedyblook".

[@catdot3889]

what are the difference between banach spaces and hilbert spaces?

[@VinVin21969]

all hilbert space are also part of banach space. but not all banach is part of hilbert

[@Jagarekca.Bahureksa]

I didn't know u r physics man

[@abcdef2069]

it cant be all the property of regular vector space that is commutable. hilbert is not commutive. what do you mean by hilbert space seperable? antilinear sounds like non-linear to me, it looks like complex conjugated linear. it seems to me the hilbert space is a complex number vector space and the normal vector space we know is a real number vector space.

[@TavartDukod]

abc def no, vector spaces can be defined over every field(it's an algebraic system, look at wiki). But hilbert spaces can be defined over both the fields of real and complex numbers. And yes, in case of spaces over the field of real numbers antilinearity is simply linearity.

[@columbus8myhw]

Co-chy! Not cow-chy! Love the videos

[@tandalkran883]

It has been almost 6 years but in case if you are still answering questions: By looking at various resources I have found out that some people say that the inner product is linear with respect to 1st vector and antilinear w.r.t 2nd vector and others say the same as you. So this linearity in the first or second vector is some kind of an optional thing, or those other people (who say 1st vector is linear) are simply wrong? Thanks btw your videos are extremely efficient.

[@DB-nl9xw]

Explain: What's the point of a Hilbert Space?

[@FacultyofKhan]

It's got certain mathematical properties (i.e. the ones discussed in the video) that come quite in handy when doing calculations in Quantum Mechanics. Maybe this might add more information: en.wikipedia.org/wiki/Hilbert_space#Applications

[@DarkPhant0m13]

riiiiight 0.o

[@GilesMcRiker]

This isn't teaching. This is just a rapid recitation of formalism. May as well just listen to an audio version of a Wikipedia page.

[@physicsstudent6313]

Tons of statements, but no demonstrations to fortify learning. Too much like just reading off a page.

[@exoendo]

ohhhh now i get it.... ._.

[@Saimouer]

Why do you use pi as an example of an element of the set of all rational numbers? Extremely confusing and not accurate.

[@mutex1024]

He uses it as an example of an irrational numeral that you can get arbitrarily close to with a rational number.

[@mutex1024]

To be honest, he messes up explaining what dense means. He should have said something like "rational numbers are dense because you can find a rational number that is arbitrarily close to any real (rational or irrational) number.

[@Jagarekca.Bahureksa]

I bet axl too 🙄🙄😭😁

[@appemail2157]

.

[@ashleytippit9500]

Uh?man,i tried....

[@andrei-un3yr]

Just looking at all those properties that define a vector space/Hilbert space makes me wonder just...why. What should I do with this information? Why do I have to learn something that seems so arbitrary

[@Adraria8]

Andrei Understanding the properties of a general Hilbert space is useful in quantum mechanics because the wave functions belong to a certain type of Hilbert space that’s called the state space

[@andrei-un3yr]

@@Adraria8 well yeah but how can I appreciate its usefulness when I start with some abstract math definitions. It would've been much more interesting to start with the properties of the space belonging to wave functions and there afterwards develop the math for it

[@Adraria8]

Andrei totally agreed!

[@ArthurHau]

The fact that Q is dense is either tautological or philosophically dangerous (or philosophically wrong)! The statement: "You can find a rational number that is as close to any irrational number as you want" is philosophically impossible because "choosing" and "finding" is by definition "countable"! You can choose only countably many times and find countably many times. Say I choose 1 time, 2 times, 3 times, ... But irrational numbers are uncountable. Or, what do you mean by "you can choose uncountably many times"?? Also, you have only countably many rational numbers to choose from, how can you fill uncountably many holes created by irrational numbers?? So, Q is dense in R is either an unproven Axiom, just like the Axiom of Choice. Or it is philosophical wrong if we believe that you can fill uncountably many holes with countably many numbers.

[@chasebender7473]

All axioms are unproven. I like the skepticism and lust for a logically complete system of mathematics, but the axiom of choice is ironically a choice. It is undecidable, and thus a consistent system can be built with or with out it. If you do not choice to accept it, that is fine, but in doing so you are throwing out so much of modern mathematics that it seems counter productive. If you do choose to choose to disregard it i would recommend you the youtube channel njwildberger, and his sister channel Wild Egg mathematics courses. This guy is a bit of a character and very adamantly opposes the axiom of choice as well as the (i believe) the axiom of infinity.

[@CstriderNNS]

WHAT ARE SOME EXAMPLES AND USES OF NON HILBERT SPACES, AND IF SPACE TIME IS QUANTISED (PLANK LENGTH) DOESN'T THIS MEAN THAT REALITY EXIST OUTSIDE HILBERT SPACE,? ops caps lol

[@chasebender7473]

Perhaps. However, Hilbert spaces are used to measure and represent the quantum states of particles which, as far as we know(don't hold me to this), do have a continuous wave function.

[@CstriderNNS]

@@chasebender7473 i wounder what information can be teased out of a system that represents the wavefunction as noncontinuous system. i.e. once quantization of the equations of BlackBodyRafiation , thw ultraviolet catastrophe was solved . I personally dont have write the mathematical ability to rewrite the equations. but i would LOVE to collaborate with some one who does

[@aakashdewangan7313]

very bad explanation.......

[@kenkal1829]

no has and no one ever will find the value of pi!!!

[@phaecops]

If you don't explain Hilbert spaces with concrete examples using real numbers instead of symbols you are failing miserably to explain it. You seem to not understand Hilbert space yourself but just regurgitate the familiar phrases and formulas that we can get from math books everywhere. Here's a suggestion. Show how a Hilbert space can model vibrating spring harmonics as a point in a hilbert space. Can you do that? I didn't think so because the people making this video just record an artificial voice reading from a textbook. The Kahn brand is ruined.

[@Paraselene_Tao]

what the fuck am I watching

[@iknownothing35]

explanation was great but the way you use your bra-ket, kinda throws me off sometimes