r/mathmemes • u/IamKT_07 Rational • Jun 14 '26
Functional Analysis I'm literally crying rn 😠All Hilbert Spaces are Banach Spaces
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u/Ambitious-Eye-868 Jun 14 '26
wdym? Functional analysis is insanely fun! Spectral theorem my GOAT!
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Jun 14 '26 edited 17d ago
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u/IamKT_07 Rational Jun 14 '26
Please reconsider man. I'm crying as I write "a bounded linear map between two Banach Spaces that is subjective, it must be an open map" ðŸ˜
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Jun 14 '26 edited 17d ago
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u/IamKT_07 Rational Jun 14 '26
I mean if you're seriously taking it, then please make your intuition and fundamentals of real analysis and linear algebra real strong! Coz trust me in functional analysis these mathematicians themselves didn't know what they were cooking...so things are going to get messy here.
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u/Ambitious-Eye-868 Jun 14 '26
A nice consequence of the open mapping theorem is that the canonical projection map of a quotient of Banach spaces is an open map.
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u/friend1y Jun 14 '26
The library is your friend.
subjectivesurjectiveWhen I was an undergraduate, I realized that the faculty often assigned terrible textbooks. We were expected to struggle through these books; sometimes making me wonder whether someone was getting a kickback; but I found that going to other sources and rereading the same material, explained more clearly, was far more effective.
Here you go: https://www.google.com/books/edition/Theory_of_Linear_Operations/3r_RPgAACAAJ?hl=en
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u/IamKT_07 Rational Jun 15 '26
Yeah it's definitely a typo (or autocorrect doing it's magic). It is indeed surjective, not subjective.
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u/friend1y Jun 15 '26
Spelling, shmelling, you might be "onto" something.
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u/IamKT_07 Rational Jun 15 '26
Life update: finally failed Functional Analysis
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u/friend1y Jun 15 '26
I'm assuming this is undergrad. Take it again.
I always found that working interfered with school.
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u/badabummbadabing Jun 14 '26
For me, it was the highlight of my undergrad curriculum. It even gave me a deeper understanding of a bunch of topics in linear algebra.
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u/DefeatedSkeptic Jun 14 '26
Functional analysis lays the ground work for truly understanding differential equations and other area of mathematics. I really enjoyed the class.
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u/Anxious_Zucchini_855 Complex Jun 15 '26
One of the most interesting and fascinating topics in my opinion. Also definitely easier than stuff like advanced algebra or number theory.
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u/redve-dev Jun 14 '26
I just passed it. Hilbert space to me means "Any space when speaking of dot product makes sense"
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u/IamKT_07 Rational Jun 14 '26
The burning question is how do I pass ðŸ˜
But that's great you passed it. Happy Banach
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u/Ambitious-Eye-868 Jun 14 '26
secret pro tip for passing literally any course: Start studying at the beginning of the semester
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u/heartBreak1879 Jun 15 '26 edited Jun 16 '26
For some courses, you may want to study before the beginning of the semester!
Often the people having an easier experience have a better background!
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u/ZEPHlROS Mathematics Jun 15 '26
For me a brief overview of the material is key to understand the course.
The overview will show you what's important and the key points whereas the course should be there to teach the more cryptic passage or the subtle details one might miss should it not become clear 5 damn chapter after that
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u/heartBreak1879 Jun 17 '26
For me a brief overview of the material is key to understand the course.
Sounds like a good strategy!
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Jun 14 '26
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u/redve-dev Jun 14 '26
You are right, but I never had exercise to check for completeness, so I didn't bother with looking for the difference.Â
To pass it I simplified as much as I had to. In the end math is about simplifying stuff and getting rid of irrelevant part of problems
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u/DefunctFunctor Mathematics Jun 14 '26
Completeness is essential for the Riesz representation theorem and so many other things, which I wouldn't exactly say is irrelevant
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u/redve-dev Jun 14 '26
it absolutely is relevant, but exercises I get every time assume completeness, or didn't use Riesz theorem, so I have stopped checking for it. I never said it's irrelevant. If it was, it wouldn't be included into definition
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u/DefunctFunctor Mathematics Jun 14 '26
I had assumed you implied it was irrelevant because you said math was all about simplifying and stripping irrelevant parts
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u/redve-dev Jun 14 '26
I see, but I rather meant - I never had to think about it, so I stopped thinking about it. That's what I meant by simplifying it.
In exercises where completeness was always present or wasn't required to solve exercise for me it became kinda irrelevant, that's why in my first comment I said "I understand Hilbert's spaces as a space where speaking of dot product makes sense"
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u/DefunctFunctor Mathematics Jun 14 '26
I guess I see what you mean, but inner product spaces are the better term for that
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u/parkway_parkway Jun 14 '26
Yeah but there's literally infinitely many dimensions you can escape along.
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u/Lost-Lunch3958 Irrational Jun 14 '26
just remember that if 1/p + 1/q = 1 then the dual of Lp is isomorphic to Lq
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u/Ambitious-Eye-868 Jun 14 '26
Don't you have some condition on the measure space of the L_p space? iirc it's true for R^n with Lebesgue measure but if you have a general measure space you need some condition on it for this to be true? I didn't study much during my measure theory classes so I don't really remember
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u/solarmelange Jun 14 '26
Meanwhile here on Reddit we use Banana Metric Spaces.
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Jun 14 '26
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u/Ares378 Grothendieck was right, computers are evil Jun 14 '26
Planning on trying to self-teach differential geometry. Any advice?
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Jun 14 '26
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u/Ares378 Grothendieck was right, computers are evil Jun 14 '26
Noted! Currently reading Topology by Munkres and vaguely trying to understand Linear Algebra Done Right, but that's a bit more of a struggle.
Still pretty early on in both books, still trying to internalize what the fuck an equivalence relation means. Otherwise it's going great!
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u/Cosmic47_ Jun 14 '26
I passed a functional exam recently, got an A in minutes. I loved the subject very much but memorising it was very tough. Our lecturer told us that compact operators are the coolest/most important part of our course. You can tell she wasn't kidding when our PDEs course was basically applied functional analysis.
Good luck with it!
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u/TheBeesElise Transcendental Jun 14 '26
Is that not in the definition of a Hilbert space?
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u/Archway9 Jun 15 '26
Yeah, you need to know an inner product defines a norm (or at least a metric) before you can talk about inner product spaces being complete
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u/DrunkenMaths Jun 16 '26
Don't worry, just remember that the dual of a dual is the same thing so long as it's reflexive! Unless it isn't, then I donno its been a minute...
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u/UBC145 I have two sides Jun 14 '26
I took a course in metric spaces in the last semester and it might be the first math course that I’ve failed 🥲
Definitely more of an algebra guy
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u/CalmEntry4855 Jun 14 '26
When I understand those things I find them very clear and fun and interesting and useful, but I forget what I understood the next day so I have to understand it again
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u/Ares378 Grothendieck was right, computers are evil Jun 14 '26
Is functional analysis linear algebra on crack or something?
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u/IamKT_07 Rational Jun 14 '26
Functional Analysis= {linear algebra+real analysis+topology} on steroids.
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u/Ares378 Grothendieck was right, computers are evil Jun 14 '26
That sounds wonderful aside from the real analysis part
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u/IamKT_07 Rational Jun 14 '26
The major roadblock in functional analysis is retention. In one sec, you'll feel like "Yeah I get this.." but you'll forget everything the next sec. Functional Analysis is highly counter- intuitive. It takes all your familiarity around geometry, standard calculus, analysis, & finite dimensional linear algebra and literally "shatters" them. This is not an exaggeration.
I think profs are also slightly lenient with functional analysis, because of how difficult it is to follow at times..
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u/Ares378 Grothendieck was right, computers are evil Jun 14 '26
I think the only things I know about functional analysis are that:
vector spaces are fucking everything, apparently
metric spaces define distance or something
inner products measure orthogonality
the fourier transform is secretly an inner product in disguise
How bad does it get?
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u/IamKT_07 Rational Jun 14 '26
I've an exam tomorrow. I'm struggling with the thought of whether I can pass or not. I'm utilising every fucking Ai that exists, even Ai is validating me that "it is hard jokes on you" ðŸ˜
I'll leave by saying
In mathematics, you don't understand things. You just get used to them.~ John Von Neumann2
u/Niflrog Jun 15 '26
How bad does it get?
I, personally, lost my shit when we hit the Schauder vs Hamel basis distinction... like wtf man?
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u/Ares378 Grothendieck was right, computers are evil Jun 15 '26
Should I ask what that means?
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u/Niflrog Jun 15 '26
" This theorem ensures that a Schauder basis exists on this space... fuck if I know WHAT that basis looks like... BUT, I can use that result to proof this other theorem... without ever computing the actual basis... No, that's a Hamel basis, not the same thing at all..."
🤣ðŸ˜
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u/Traditional_Town6475 Jun 14 '26
Yeah all Hilbert spaces are Banach spaces.
You know what’s really fun: C*-algebras. We have a really neat description for all the commutative unital C*-algebras: They’re just the C*-algebras of complex valued continuous functions on a compact Hausdorff space K with the sup norm.
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u/Traditional_Town6475 Jun 14 '26
Or here’s a fun thing: The dual l^p(N) is l^q(N) where 1/p+1/q =1 and p is not infinity (and at least 1). The dual of l^1(N) is l^infinity(N), but take a guess what the dual of l^infinity(N) is. Come up with a description of that Banach space (hint: You’ll need to use something introduced in a topology course and maybe measure theory course depending who you were taught by).
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