Give me your most powerful mathematical theorem...

Give me your most powerful mathematical theorem. The first time you read it you were bedazzled; it can't be true! The statement was just too grand! But no it was true.

I think overwhelming majority of mathematical theorems are idiotic, too simple, pointless, uneconomic, nonproductive, obvious, discuss some meaningless concept like philosophy or infinity.

But here are few I actually find very useful:
1. Pythagorean theorem
2. General rule of derivation
3. Four color theorem
4. Taylor's theorem
5. Binomial theorem
6. Fourier series
7. Dirac delta function
8. Brouwer fixed point theorem
9. Arrows impossibility theorem

Noether's theorem.

Yeah this one is nice too

Absolute value lemma blew my mind and load.

Like wow. |x| < y -y < x < y. That's just beautiful. I always shed a tear.

All right but this is obvious to anyone with a brain

It says that you have two lines X and Y. X is shorter than Y.

No shit, it's a meme """lemma""" to teach brainlets how to prove triangle ineq in baby analysis. It's just the definition of AV written in such a way that even brainlets can see the connection.

[math]\exists y\forall x, p(y)\implies p(x)[/math]
it was formally proven in a logic class,
kinda like that there is a formal proof that we can prove something for an arbitrary y and then infer that it holds for all x.

that one about conditionally convergent series

[math] \int _{\partial \Omega} \omega = \int _{\Omega} d\omega[/math]

meanignless unless you defined what the fuck those 4 symbols mean

just like
abcd = fg

Yeah but everyone here knows what it means.

Explain to a chemistry brainlet?

I am a mathematician myself but nothing I've learnt has come close to the Noether theorem. It is so simple but when you start reading QFT you realise just how important symmetry is. I think it should be at least mentioned in schools.

I also remember being excited by the Gleason theorem (on density operators) and the Frobenius theorem (on R/C/H algebras) in uni. More recently, I can finally appreciate the beauty of the Atiyah–Singer theorem on analytical/topological indices.

>Noether's theorem
>Noether
>No ether

Fucking anti-ether fags still trying. Ether will prevail.

It's stoke's theorem, look it up. lots of applications

>Gödel's completeness theorem
Huh, thats really interesting I guess
>Gödel's incomplentess theorem
OVER THE LINE

>the inverse function theorem and the chain rule
hands down, those are the two theorems making any theory involving differentiable manifolds possible

1+2+3+4+5+.... = -1/12

The well-ordering theorem. This transfinite stuff still looks a bit mathemagic to me.

cantor's diagonal

>inverse function theorem
>not Implicit function theorem which has the inverse function theorem as a corollary

The pigeon hole theorem.

If you have N holes... And N+1 pigeons...

...you cannot give own hole to each pigeon...

MIND = BLOWN

...

OP posts bait thread.
But nobody falls for it.
I'm sorry OP.

Gij, j=0

they are completely equivalent. you can prove implicit f.t. from inverse f.t. easily

>Dold-Kan Correspondence
>Brown's representability theorem
Literally jaw dropping

>doesn't understand standard notation
>"muh meaningless!"

Cauchy's residue theorem is pretty neat, I guess.

Cij= (-1)^i+j * Mij

1+1/2+1/4+1/8+.... =/= infinity

that's wrong though

>residues

Complex analysis was a mistake

Not a theorem, but a principle from Linear Algebra. Shakes me to my core every time.

Every invertible linear transformation is just a change of basis matrix.

if you're studying math or physics, wait till you get to differential geometry, principal bundles stuff. shit's gonna blow your mind.

von neumann ergodic theorem

Doubtful. Those both read as weak. The linear thing really changed how I see physics and the universe. Principal bundles are... just things. Nothing follows, it's a definition.

There are interesting theorems for principle bundles.

For instance G-bundles w/ flat connection (over a space X) are in bijection with homomorphisms from the fundamental group of X to G.

I'm sure you can pick some physical significance out of that, considering gauge fields are connections on principle bundles.

>I'm sure you can pick some physical significance out of that
>I'm sure

meaning you haven't yet
meaning its a little bit less meaningful

I'm not a physicist, I know that what a physicist calls a gauge field is a connection on a principle bundle. But I don't know anything about the physics of gauge fields.

You can use them to calculate crazy integrals that you normally can't

That shit is beautiful. I came in the back of some chicks head in class when we went over it.

That's my point. You don't need to know anything about physics to understand the meaning of what I said. Knowing and understanding the math is the important part. Any linear transformation, ANY LINEAR TRANSFORMATION is just a change of basis matrix. Think about it. Push a car across a road? It never even moved. We just changed some perspective.

you're a dumbass. you don't realize the person you're speaking to knows and understands much more about all this than you'll ever do. you're saying meaningless shit.

Nullstellensatz

Kodaira Embedding

GAGA

t. highschool brainlet who browses this board thinking it makes him look smart

>Any linear transformation, ANY LINEAR TRANSFORMATION is just a change of basis matrix. Think about it. Push a car across a road? It never even moved. We just changed some perspective.
this logic is kind of misleading

brought me back to the old futurama episode where the ship moves the universe

DUDE, I bet you watch Rick and Morty, am I right?

a transformation and a change of coordinates are two very different things. just because they coincide in the very special case of R^n, it doesn't mean that they can be considered the same concept.

Ito's lemma
spectral theorem
Feynman-Kac formula

No, they aren't. So long as it's linear, and so long as it's invertible. Which I'll grant is pretty limiting, but hey. It's what OP was asking for.

>only having read one approach

Yoneda Lemma
Donaldson's Theorem

Patricians

Look at me I don't know what a manifold is, this is meaningless

kek

thats not wrong. just dont put that in your math or calculus test.

HNNNNNNNNG

One of my favourites is Weierstrass Factorization Theorem for Entire Functions.

My man Euler just assumed it's true cause it is too natural not to be.

Actually has some really cool applications if you take the time to read a combinatorics book

Euclid's Division Lemma

Godel's proof restored my faith in math

Brainlet

>the CLiT theorem

I like this painting. Who's the artist?

Rao-Cramer bound, Stein's paradox.

same can be said of any theorem