## Fundamental theorems [closed]

I am certainly sure that any one who has read Gil Kalai's witty community wiki has benefited a lot. Here I follow a similar track in asking this question. So let's compose a list of fundamental theorems in mathematics which may not even have the tag "fundamental" but have serious wight in the respective branch of math.

I will start with the elementary and very popular ones.(Please add a description if the theorem is fundamental but still not so well-known)

Thanks for all your effort.

1. FTA: The Fundamental Theorem of Arithmetic (or Unique-Prime-Factorization Theorem): ->Any integer greater than 1 can be written as a unique product (up to ordering of the factors) of prime numbers.

2. FTA: The Fundamental theorem of Algebra: -> The field of complex numbers is algebraically closed

3. FTC: The fundamental theorem of calculus: -> Has two parts and specifies the relationship between the two central operations of calculus: differentiation and integration.

4. FTLP: The fundamental theorem of linear programming: -> In a weak formulation, states that the maxima and minima of a linear function over a convex polygonal region occur at the region's corners.

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I don't really understand what distinguishes a "fundamental theorem" from an important theorem apart from having the luck to have been called a fundamental theorem. Is it really true, for instance, that the FTC is really the most important theorem in calculus? Maybe it's the most important theorem in highschool calculus, but, I would say that there are a large number of theorems of real analysis in one variable that are as important if not moreso. Similarly, is the fundamental theorem of algebra really important for algebra, or should it be the fundamental theorem of complex numbers? – Harry Gindi May 20 2010 at 8:35
I don't know how standard they are, but a Google search reveals that at least some people refer to fundamental theorems of Galois theory, space curves, projective geometry and Riemannian geometry. – gowers May 20 2010 at 9:22
Okay we may stay content with those that have the tag only or what people consider fundamental but not tagged that way. – To be cont'd May 20 2010 at 9:24
I voted to close. I see no commonality between answers to this question other than a linguistic one (ie someone decided to call a lemma "fundamental"). – Andy Putman May 20 2010 at 18:34
Closed. I'm not excited about jumble-bag big-list questions, and I think this one is not going to be very useful to anyone, so I've closed it. Wikipedia seems to like big lists, and I'd encourage anyone excited about this particular list to try it out there. – Scott Morrison May 20 2010 at 19:04
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## 5 Answers

In his book Topics in Geometric Group Theory, Pierre de la Harpe calls the following result the Fundamental Observation of Geometric Group Theory (though he also calls it a theorem!). It is also often called the Svarc--Milnor Lemma. Roughly speaking, it asserts that the coarse geometry of a group is captured by any suitably nice action of that group by isometries on a metric space.

Theorem. Let $X$ be a metric space that is geodesic and proper, let $\Gamma$ be a group and let $\Gamma$ act properly discontinuously and cocompactly by isometries on $X$. Then $\Gamma$ is finitely generated, and furthermore for any $x_0\in X$ the map $\Gamma\to X$ given by

$\gamma\mapsto\gamma x_0$

is a quasi-isometry.

Remarks.

1. $\Gamma$ is endowed with the word metric (with respect to some choice of finite generating set).
2. A map of metric space $f:Y\to X$ is a quasi-isometric embedding if there are constants $\lambda\geq 1$, $\mu\geq 0$ such that

$\lambda d_Y(y_1,y_2)+\mu\geq d_X(f(y_1),f(y_2))\geq \frac{1}{\lambda} d_Y(y_1,y_2)-\mu$

for all $y_1,y_2\in Y$. It is a quasi-isometry if, furthermore, for every $x\in X$ there is $y\in Y$ such that $d(x,f(y))\leq \mu$.

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I used to joke that The Fundamental Theorem of Combinatorics is interchange of summation.

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 See also: en.wikipedia.org/wiki/… – Douglas S. Stones May 21 2010 at 12:15

To add to the Gowers examples: the fundamental theorem on finitely-generated abelian groups. It seems at least a mildly interesting linguistic point. German discriminates between Hauptsatz and Fundamentalsatz, i.e. main theorem and fundamental theorem (if satz is not quite "theorem"). That distinction seems less clear in the English usage. The German Wikipedia admits the Fundamental Theorems of Algebra, Analysis and Arithmetic, but others in pure mathematics aren't obvious. I would myself think of Galois theory (the perfect duality of subfields and subgroups) and projective geometry (collineations semi-coordinatised) as having "fundamental theorems".

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Welcome to MO, Charles! – Artie Prendergast-Smith May 20 2010 at 12:31
Thanks. But I'm really a double agent, working for you-know-who. – Charles Matthews May 20 2010 at 14:42

The Fundamental Theorem of Asset Pricing (FTAP) in mathematical finance also comes in two parts. The first part says, more or less, that a market is arbitrage-free if and only if there is an equivalent martingale measure for the discounted price process. The second part says that the market is complete (all European options can be hedged) if and only if the equivalent martingale measure is unique.

(In some models, you may need an appropriate definition of "arbitrage-free", such as the notion of "no free lunch with vanishing risk", and you may replace "equivalent martingale measure" with "equivalent local martingale measure". But the idea is the same.)

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Wikipedia says the Fundamental Theorem of Riemannian Geometry is the unique existence of the Levi-Civita connection. I've never heard it called that myself, so this is maybe an anti-answer.

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