/sci/ - Science & Math

[Anonymous No.14861627]

Functional analysis is the most chad field of maths out there. It strikes the perfect balance between being theoretical and applicable, while being fundamental at the same time.

[Anonymous No.14861634]

>>14861627
All math is Abstract Algebra.

[Anonymous No.14861643]

>>14861627
Any field which uses "real numbers" cannot possibly be fundamental
>Verification not required

[Anonymous No.14861665]

>>14861634
>>14861643
YWNBAR Mathematician.

[Anonymous No.14861936]

>>14861634

[Anonymous No.14862533]

Based. Fun anal is studied best in the context of PDE thoughever. In isolation you'd just get a dry intro to Hahn-Banach, open mapping and uniform boundedness.

[Anonymous No.14862604]

>>14861627
Next to differential geometry it's the most useful math for use in physics you can't convince me otherwise

[Anonymous No.14862629]

>>14862604
ode pdes special functions complex analysis?

[Anonymous No.14862653]

>>14862629
Complex analysis is definitely a close third

pde/ode/special functions is useful to know but 9 times out of 10 you only need to know enough to look up properties or do numerical work.

[Anonymous No.14862675]

>>14862533
It's needed for all the fun stuff (as in fun anal), kind of like Lebesgue Integration theory. Although Fun anal also needs lebesgue integration

[Anonymous No.14862677]

>>14862604
Name one use of functional analysis for physicists. No I don't mean "mathematical physicists".

[Anonymous No.14862680]

>>14862677
Quantum mechanics

[Anonymous No.14862688]

>>14862604
>>14862677
For reference
>In the 1960s Friedrichs met
Heisenberg, and used the occasion to express to him the deep gratitude of the
community of mathematicians for having created quantum mechanics, which gave birth
to the beautiful theory of operators in Hilbert space. Heisenberg allowed that this was
so; Friedrichs then added that the mathematicians have, in some measure, returned
the favor. Heisenberg looked noncommittal, so Friedrichs pointed out that it was
a mathematician, von Neumann, who clarified the difference between a self-adjoint
operator and one that is merely symmetric. "What's the difference," said Heisenberg.

[Anonymous No.14862691]

>>14862677
hydrodynamics, stat mech, qft

[Anonymous No.14862692]

>>14862680
See >>14862688.

[Anonymous No.14862694]

>>14862691
>hydrodynamics, stat mech, qft
None of those use functional analysis

[Anonymous No.14862783]

>>14862675
>Fun anal
>>>/lgbt/
Go back to your containment board and don't promote your degenerate lifestyle here.

[Anonymous No.14862908]

is Abbott's Understanding Analysis enough to prepare you for fun anal?

[Anonymous No.14863075]

>>14862908
nah you needs 2 real analysis courses, complex analysis, abstract algebra and general topology

[Anonymous No.14863293]

>>14863075
Is the OP book a good one? Is Amann & Escher good enough preparation? What complex analysis would you need?

[Anonymous No.14863309]

>>14863293
Amann Escher + some basic linear algebra is all you need before starting fun anal.

[Anonymous No.14863609]

>>14862783
>

[Anonymous No.14863974]

as an engineer should I study functional analysis??

[Anonymous No.14863978]

>>14863974
>as an engineer
You should learn it as person in pursuit of knowledge.

[Anonymous No.14863989]

>>14863974
Are you involved in developing novel numerical solutions to engineering problems? Im assuming that answer will be no, as I'm also a PE and the only times I've heard terms like functional analysis thrown about, have been in conjunction with numerical methods and analysis. I do not know how they relate, but its what the more "mathy" coworkers talk about about.

[Anonymous No.14864037]

>>14862783
>here comes a faggot

[Anonymous No.14864089]

>>14863974
Functional analysis is analysis but of functionals, which are functions where the input is a function. Calculus of variations is the motivating part of it. The oldest example is of course the Brachistochrone curve, where you have to find right path in form of a function that minimise time. Now you figure.

[Anonymous No.14864178]

>>14862688
Friedrichs was right but his reasoning was wrong. How does distinguishing between self-adjoint and symmetric operator constitute some sort of great discovery? It's not a discovery. It's just a definition. You can make your definitions as specific or as general as you'd like. If you make your definitions more specific, so that you can distinguish between symmetric operators, in general, and self-adjoint operators in particular, that can certainly be a useful distinction to make (especially in the complex vector spaces), but it doesn't constitute a discovery or a new insight in its own right.

I would definitely be willing to view a theorem or a lemma or a corollary as a discovery, but a definition or a collection of definitions is not a discovery. It's more like an arbitrary stipulation than a discovery. We can use definitions to help us understand things and generate new theorems and other results that genuinely do constitute discoveries, but in itself, the act of distinguishing between symmetric and self-adjoint operators is not really a discovery, it's just a convention for classifying certain linear operators.

If you actually want to talk about practical uses of functional analysis, I would say look to it's connections with lie groups and representation theory on the one hand, and then with pdes and dynamical systems on the other.

[Anonymous No.14864183]

>>14864178
The fact that you posted that picture makes me not want to read your post. Try again without bringing your shitty politics into it.

[Anonymous No.14864248]

>>14864089
That is only partially true. The naming 'Functional Analysis' is a bit misleading. FA is an extremely broad field and functionals are only a slim part of it. Dieudonne History of analysis has more on it.

[Anonymous No.14864291]

>>14864183
Don't worry, you probably don't have enough background knowledge in math or physics to even understand my post, so it doesn't really matter.

Anyway, I find it funny that woke, pro-censorship leftists feel comfortable pushing their politics on other people and demanding that they conform to their political values, but when people from literally any other political ideology other than neoliberalism (even other leftists) vocalize their own positions, that's absolutely unacceptable, and they're "politicizing" things.

For instance, in recent years woke neoliberal like yourself - working in lockstep with neocon christians and zionists - have enacted laws in over 30 US states that prohibit public employees, such as teachers and state university professors, from promoting Palestinian self-determination and the BDS movement. So how come when it comes to Palestine and Zionism, woke neoliberal have no problems "politicizing" education or academia?

How come when it comes to the Lab Leak Hypothesis you people had no problem politicizing things and falsely claiming, without evidence, that the Lab Leak was an anti-chinese white nationalist conspiracy theory? Don't you think that was politicizing the issue a bit?

[Anonymous No.14864325]

>>14864291
>>14864178
>woke
>neoliberal
>Israel
>Zionism
>Lab Leak

So many poltard buzzwords, and yet not a single coherent thought was expressed in either of these posts. Incels and poltards always try to make themselves out to be a victim. Poltards and right winger are not victims. Islamic terrorists attacking Israel or America are not victims. Russia and Valdimir Putin invading Ukraine is not a victim. Taking a contrarian political position on every single political issue under the sun does not make you clever, and it doesn't make you a "free thinker" who is above all the "sheeple". You're a contrarian edgelord, and like all contrarian edgelords your political and social views are largely rooted in a deep seated sense of spite and hatred and inferiority. Maybe if you weren't a whiney right wing incel constantly complaining about the "normies" and the government and the media and the liberals, and literally anyone and everything in society but yourself, then people wouldn't be so annoyed and off-put by you and other poltards like yourself, but I guess basic self-awareness is hard to come by amongst autistic poltards and 4chan incels.

[Anonymous No.14864339]

>>14864325

[Anonymous No.14864355]

>>14864291
>you probably don't have enough background knowledge in math or physics to even understand my post,
I know more than you as seen by your cluelessness in the second sentence of your earlier post. To Friedrichs, a mathematician, the difference between symmetric and self-adjoint operators is clearly important as you can only apply the spectral theorem to the latter. To physicists, all of this implicit in the way they talk about the boundary conditions of the wavefunctions they consider so it's not particularly new.

[Anonymous No.14864392]

>>14864355
>To Friedrichs, a mathematician, the difference between symmetric and self-adjoint operators is clearly important as you can only apply the spectral theorem to the latter. To physicists, all of this implicit in the way they talk about the boundary conditions of the wavefunctions they consider so it's not particularly new.

Yes, none of this is news to me, which is why I said
>You can make your definitions as specific or as general as you'd like. If you make your definitions more specific, so that you can distinguish between symmetric operators, in general, and self-adjoint operators in particular, that can certainly be a useful distinction to make (especially in the complex vector spaces)

I understand the distinction between self-adjoint and symmetric operators, and I even indicated as much in my post -- and I'm actually taking a grad course on differential galois theory right now, and it's all about operators over complex projective spaces, where all of this stuff is relevant. I'm not saying that self-adjoint and symmetric operators are the same or that the distinction is useless. My point is that distinguishing between them is not a discovery, it is a definition.

This should not even be a controversial claim. You've obviously not studied much logic or model theory, which is all about the distinction between definability and provability. This is more of a philosophical point than a mathematical one, but the definitions relate to the logical structure of the theory in question, i.e. the linguistic and axiomatic framework in which claims (i.e. theorems, lemma, etc.) can be proven. Definitions are not discoveries. They are the starting point for discoveries.

Also, you still haven't addressed why neoliberal centrists should get to censor and impose their views on others, and why conservatives and leftists should not be allowed to do the same.

[Anonymous No.14864421]

>>14864392
>Yes, none of this is news to me, which is why I said
Your mention of complex vector spaces was off to me since it should have been obvious that we were talking about hilbert spaces in quantum mechanics. You would have sounded more knowledgeable if you had said "in infinite dimensional hilbert spaces" instead of "especially complex vector spaces".
>you still haven't addressed why neoliberal centrists should get to censor and impose their views on others
Never said that. I just didn't want to talk with you since you just couldn't contain your transphobia to yourself

[Anonymous No.14864446]

>>14864355
>>14864392
Just to elaborate on why I say that definition are not discoveries, let me explain. First of all definitions are arbitrary and their are often many distinct but equally strong ways of defining the objects of some mathematical theory.

When I say that providing a list of definitions does not constitute a discovery, I do not mean that as a value judgement. I'm not saying that providing definitions is useless or a waste of time. It's just a logically distinct process from deriving new theorems. Devising new definitions and axiom systems is analogous to the process of devising new experimental methods in the experimental sciences. If you develop a new experimental method, that's not a discovery. That's not to say it isn't useful or insightful or scientifically valuable. You haven't produce a new discovery, you've produce a new methodological framework in which discoveries can be made.

For instance, and this is a trivial example, but I could define two new classes of graphs. Graphs whose vertex sets are sizes which are multiples of the number 17 and graphs with any other number of vertices. Call the former family of graphs A and the latter family B. At this point, I have not made any new discoveries, because we don't know anything in addition about graphs. All I have done is several families of graphs. However, if I then go on to prove additional theorems about each of these respective families of graphs, that would constitute a discovery. You can create whatever definitions you'd like, but without additional theorems or lemma or results on top of that, it does not constitute a discovery. For instance, one of the things that makes the difference between self-adjoint and symmetric operators interesting, is precisely because over complex vector spaces, the former admit a spectral decomposition, whereas the latter do not. This has a lot of consequence, both computationally, but also in terms of representation theory and decomposing representations.

[Anonymous No.14864486]

>>14864421
>Your mention of complex vector spaces was off to me since it should have been obvious that we were talking about hilbert spaces in quantum mechanics. You would have sounded more knowledgeable if you had said "in infinite dimensional hilbert spaces" instead of "especially complex vector spaces".

You probably know more physics that I do, so you're used to working with hilbert spaces and quantum mechanics. The reason I mention complex space, is because in math (which is what I'm used to) the distinction between complex and real vector spaces is very important, and when you study representation theory (which is basically the study of matrices acting on vector spaces), a representation of a group is irreducible iff the associated matrices are diagonalizable. This is equivalent to saying the are self-adjoint. Over real vector spaces, this is overkill, since any symmetric matrix is self-adjoint.

[Anonymous No.14864546]

>>14864486
I'm well aware of representation theory (which also comes up a lot in physics). Your definition of irreducibility is not really correct however. Also, in finite dimensional complex inner product spaces, self-adjoint and symmetric are interchangeable. You're probably using "symmetric" for complex matrices in the sense that it's used for real matrices but that's not really what people mean when they say symmetric in functional analysis.

[Anonymous No.14864558]

>>14864546
>>14864486
>You're probably using "symmetric" for complex matrices in the sense that it's used for real matrices
By which I mean a matrix with entries a_ij = a_ji. This is not the definition of symmetric in functional analysis that Friedrichs was referring to.

[Anonymous No.14864597]

>>14863974
You shouldn't bother unless you're going into advanced PDE's

[Anonymous No.14865708]

What are the best functional analysis books?

[Anonymous No.14865713]

>>14865708
rudin

[Anonymous No.14865719]

any use for functional analysis in computer science?

[Anonymous No.14865725]

>>14865713
In the references he lists Dunford & Schwartz's Linear Operators, what's that book like? It seems huge. Is it still relevant? It's quite old.

[Anonymous No.14865739]

>>14861627
You should learn model theory. I hate model theory, and even I can recognize that model theory is currently It™.

[Anonymous No.14865916]

>>14865713
wrong

[Anonymous No.14866417]

>>14865713
I said wrong

[Anonymous No.14866492]

>>14862677
https://arxiv.org/abs/1911.06630

[Anonymous No.14866498]

>>14862694
qft uses fun anal
>>14865708
methods of mathematical physics volumes I-IV
>>14865719
in machine learning yes

[Anonymous No.14866506]

>>14864325
>Reeeeeeeeeeeeeee incels
>Reeeeeeeeeeeeeee conservatives
>Reeeeeeeeeeeeeee conspiracy theorists
>Reeeeeeeeeeeeeee Muslims and Palestinians

It's truly amazing how contemporary mainstream liberals have managed to construct an ideology that simultaneously demonizes both conservatism and socialism and basically any ideology other than neoliberalism, while also painting themselves as """anti-racists""" despite the facts that they exhibit an almost irrational and xenophobic hatred of Iran, Palestine, Syria, and Russia, and basically any other country that posses a threat to their Zionist foreign policy and expansionist military agenda.

[Anonymous No.14867089]

>>14866506
sir this is a /sci/ence board

[Anonymous No.14868809]

>>14864392
> I'm actually taking a grad course on differential galois theory right now, and it's all about operators over complex projective spaces, where all of this stuff is relevant.
You're a LARPer desperately trying to impress midwits by using big math words but you obviously have no idea what you're talking about.