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/sci/ - Science & Math

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14628975 No.14628975 [Reply] [Original]

post your favourite kino theorems, I'll start
Hilbert Projection Theorem
>In a hilbert space, every nonempty convex set has a unique point minimizing the distance to a point outside the set
Rademacher theorem
> every real valued function defined on a real valued open subset and Lipschitz is differentiable almost everywhere

>> No.14629021

>>14628975
Something’s wrong in your Hilbert projection theorem statement.

Consider the reals, this is a Herbert space.
Consider the set of all points between but not including 0 and 1, this set is convex.
Consider the point at 1, which is not in the set.
There is not any (let along unique) point in the set (0 , 1) that attains the minimum distance between points in (0, 1) and 1; proof: assume for the sake of contradiction there is such a point x in (0,1). But (1+x)/2 is also in the set yet is closer to 1 than x is, thus contradicting the assumption. QED

>> No.14629038

>>14629021
okay, forgot the set has to be also closed, for obvious reasons.
> Herbert space

>> No.14629231

>>14628975
>almost everywhere
cope

>> No.14629805

>>14629021
The point in (0,1) closest to 1 is clearly 0.999....

>> No.14629904

27

>> No.14629907

>>14628975
cool art

>> No.14630261

>>14629231
the intersection of your mathematical abilities with the set tools needed for mentally grasping the perfectly valid theory of real numbers forms an empty set.

>> No.14631057

does /sci/ do math?

>> No.14631222

My favorites are spectral theorem, Banach-Alaoglu, and Atiyah-Singer index theorem.

>> No.14631229
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14631229

Kronecker's theorem on simultaneous approximation. Besides the theorem and its proof being very cool, ironically it's completely nonconstructive.

>> No.14631239

>>14631222
>>14631229
can you explain those a bit, what the entail and where they are used? they sound interesting

>> No.14631712
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14631712

>>14631239

>> No.14634462

>>14631712
waiting.

>> No.14634515

>>14631229
which proof? there are several of them

>> No.14634556

>>14634515
I had in mind the beautiful proof presented in Cassels book on Diophantine approximation. There is another proof? How is it different?

>> No.14634578

>>14634515
Preferably all, since I know none of them, but you can pick your favourite of those
>>14634556
stop impersonating me

>> No.14634586

>>14634578
Wtf?

>> No.14635919

>>14628975
Analysis:
>contraction mapping theorem is cute.
>newton's method for finding roots. (multivariate version.)
>the version of the existence-uniqueness theorem for ODEs which uses polygonal curves instead of a picard iteration.
Algebra:
>schur decomposition.
>spectral theorem for normal matrices
Geometry:
>grassmanian iteration methods for computation of eigenvalues.