SOME FUNDAMENTAL THEOREMS IN MATHEMATICS

1 SOME FUNDAMENTAL THEOREMS IN MATHEMATICSOLIVER expository hitchhikers guide to some THEOREMS in for the current list of 250 THEOREMS are whether the result can be formulated elegantly,whether it is beautiful or useful and whether it could serve as a guide [6] without leadingto panic. The order is not a ranking but ordered along a time-line when things were writ-ten down. Since [570] stated a mathematical theorem only becomes beautiful if presentedas a crown jewel within a context" we try sometimes to give some context. Of course, anysuch list of THEOREMS is a matter of personal preferences, taste and limitations. The num-ber of THEOREMS is arbitrary, the initial obvious goal was 42 but that number got eventuallysurpassed as it is hard to stop, once started. As a compensation, there are 42 tweetable" THEOREMS with included proofs.

2 More comments on the choice of the THEOREMS is included inan epilogue. For literature on general MATHEMATICS , see [198, 194, 29, 241, 261, 633, 422, 143],for history [223, 639, 386, 76, 48, 213, 389, 374, 706, 116, 632, 82, 266, 349], for popular,beautiful or elegant things [12, 542, 206, 187, 17, 686, 687, 46, 209, 195, 251, 456, 630, 310,206, 2, 130, 151, 131, 515, 268, 177]. For comprehensive overviews in large parts of math-ematics, [77, 170, 171, 53, 607] or predictions on developments [49]. For reflections aboutmathematics in general [150, 466, 47, 313, 449, 102, 575]. Encyclopedic source examples are[193, 722, 684, 105, 197, 157, 227, 196, 114, 649].This is a live document which is in the process of being extended. Thanks so far to JohanCommelin, Mikhail Katz, David McCarthy, Kapil Paranjape, Jordan Stoyanov, Michael Somos,Ross Rosenwald for some valuable comments or {0,1,2,3.}

3 }be the set ofnatural numbers. A numberp N,p >1isprimeifphas no factors different from1andp. With aprime factorizationn= , weunderstand the prime factorspjofnto be ordered aspi pi+1. Thefundamental theoremof arithmeticisTheorem:Everyn N,n >1has a unique prime anticipated the result. Carl Friedrich Gauss gave in 1798 the first proof in his monograph Disquisitiones Arithmeticae". Within abstract algebra, the result is the statement that thering of integersZis aunique factorization domain. For a literature source, see [363]. Formore general number theory literature, see [333, 119].Date: 7/22/2018, last update 03/02 aninner product space(V, )withdot productv wleading tolength|v|= ,three non-zero vectorsv,w,v wdefine aright angle triangleifvandwareperpendicularmeaning thatv w= 0.

4 Ifa=|v|,b=|w|,c=|v w|are the lengths of the three vectors, thenthePythagoras theoremisTheorem:a2+b2= by Babylonians mathematicians in examples, it appeared independently also inChinese MATHEMATICS [644] and might have been proven first by Pythagoras [636] but alreadyearly source express uncertainty (see [361] p. 32). The theorem is used in many parts ofmathematics like in thePerseval equalityof Fourier theory or that for uncorrelated randomvariables the variance is additiveVar[X] + Var[Y] = Var[X+Y]. In linear algebra it generalizesto theLagrange identitydet(FTF) = |P|=mdet2(FP)which holds for alln mmatrices,where the sum to the right is over allm msub-matricesPofF[32], a formula which incalculus becomes|~v|2|~w|2 (~v ~w)2=|~v ~w|2. See [366, 548, 461, 374]. a function of one variables which iscontinuously differentiable, meaning thatthelimitg(x) = limh 0[f(x+h) f(x)]/hexists at every pointxand defines a continuousfunctiong.

5 For any such functionf, we can form theintegral baf(t)dtand thederivatived/dxf(x) =f (x). theorem : baf (x)dx=f(b) f(a),ddx x0f(t)dt=f(x)Newton and Leibniz discovered the result independently, Gregory wrote down the first proofin his Geometriae Pars Universalis" of 1668. The result generalizes to higher dimensions inthe form of theGreen-Stokes-Gauss-Ostogradski theorem MdF= MFwhich holdsforn-formsFwithexterior derivativedFand compact(n+ 1)-manifoldsMwith boundary M. [202] tells the tongue in the cheek" proof: as the derivative is a limit ofquotientofdifferences, the anti-derivative must be a limit ofsumsofproducts. For history, see[373, 199]. a complex-valued function of the formf(x) =a0+a1x+ +anxn, wherethe entriesakare in the complex planeC. The space of all polynomials is denoted byC[x].

6 Thelargest non-negative integernfor whichan6= 0is called thedegreeof the polynomial. Degree1polynomials arelinear, degree2polynomials are calledquadraticetc. Thefundamentaltheorem of algebraisTheorem:Everyf C[x]of degreencan be factored intonlinear result was anticipated during the 17th century. The first author to assert that any n thdegree polynomial has a root is Peter Roth in 1600 [544]. This was proven first by Carl FriedrichGauss and finalized in 1920 by Alexander Ostrowski who fixed a topological mistake in Gaussproof. The theorem assures that the field of complex numbersCisalgebraically closed. Forhistory and many proofs see [222].2 OLIVER a sequenceXkofindependent random variableson a probability space( ,A,P)which all have the samecumulative distribution functionsFX(t) = P[X t].

7 Thenor-malized random variableX=is(X E[X])/ [X], whereE[X]is themean X( )dP( )and [X] = E[(X E[X])2]1/2is the standard deviation. A sequence of random variablesZn Zconverges in distributiontoZifFZn(t) FZ(t)for alltasn . IfZis aGaussian random variablewith zero meanE[Z] = 0and standard deviation [Z] = 1, thecentral limit theoremis: theorem :(X1+X2+ +Xn) Zin in a special case by Abraham De-Moivre for discrete random variables and then byConstantin Carath odory and Paul L vy, the theorem explains the importance and ubiquityof theGaussian density functione x2/2/ 2 defining thenormal distribution. TheGaussian distribution was first considered by Abraham de Moivre in 1738. See [638, 396]. arandom variableon aprobability space( ,A,P)for which|X|has finitemeanE[|X|].

8 This meansX: Ris measurable and |X(x)|dP(x)is finite. LetTbe anergodic, measure-preserving transformation from to .Measure preservingmeans thatP[T 1(A)] =P[A]for allmeasurable setsA that thatT(A) =AimpliesP[A] = 0orP[A] = 1for allA A. Theergodic theoremstates, that for an ergodictransformationTon has: theorem :[X(x) +X(Tx) + +X(Tn 1(x))]/n E[X]for almost theorem from 1931 is due to George Birkhoff and is calledBirkhoff s pointwise ergodictheorem. It assures that time averages" are equal to space averages". A draft of thevonNeumann mean ergodic theoremwhich appeared in 1932 by John von Neumann hasmotivated Birkhoff, but the mean ergodic version is weaker. See [721] for history. A specialcase is thelaw of large numbers, in which case the random variablesx X(Tk(x))areindependent with equal distribution (IID).

9 The theorem belongs to ergodic theory [292, 148,608]. theoryAbijectionis a map from a setXto a setYwhich isinjective:f(x) =f(y) x=yandsurjective: for everyy Y, there existsx Xwithf(x) =y. Two setsX,Yhave thesamecardinality, if there exists a bijection fromXtoY. Given a setX, thepower set2 Xis theset of all subsets ofX, including theempty setandXitself. IfXhasnelements, the powerset has2nelements. Cantor s theorem isTheorem:For any setX, the setsXand2 Xhave different result is due to Cantor. Taking forXthe natural numbers, then everyY 2 Xdefines areal number (Y) = y Y2 y [0,1]. AsYand[0,1]have the same cardinality (asdoublecounting pair = a countable set), the interval[0,1]is uncountable. There are different types of infinities leading tocountable infinite setsand3 FUNDAMENTAL THEOREMS uncountable infinite sets.

10 In order to compare sets, theSchr der-Bernstein theoremisimportant. If there exist injective functionsf:X Yandg:Y X, then there exists alsoa bijectionX Y. This result was used by Cantor already. For literature, see [293]. space( ,A,P)consists of a set , a -algebraAand aprobability mea-sureP. A -algebra is a collection of subset of which contains the empty set and whichis closed under the operations of takingcomplements,countable unionsandcountableintersections. The functionPonAtakes values in the interval[0,1], satisfiesP[ ] = 1andP[ A SA] = A SP[A]for any finite or countable setS Aof pairwise disjoint sets. Theelements inAare calledevents. Given two eventsA,BwhereBsatisfiesP[B]>0, one candefine theconditional probabilityP[A|B] = P[A B]/P[B].Bayes theoremstates: theorem :P[A|B] = P[B|A]P[A]/P[B]The setup stated theKolmogorov axiomsby Andrey Kolmogorov who wrote in 1933 the Grundbegriffe der Wahrscheinlichkeitsrechnung" [414] based on measure theory built by EmileBorel and Henry Lebesgue.