Concepts in mathematics which are referred to as 'generalizations'

Question

I am curious to know some theorems usually taught in advanced math courses which are considered 'generalizations' of theorems you learn in early university or late high school (or even late university).

For example, I know that Stokes's theorem is a generalization of the divergence theorem, the fundamental theorem of calculus and Green's theorem, among I'm sure many other notions.

I've read that pure mathematics is concerned mostly with the concept of 'generalization' and I am wondering which theorems/ideas/concepts, like Stokes's theorem, are currently celebrated 'generalizations' by mathematicians.

Answer 1

The Yoneda lemma of category theory says that every category can be understood as a category whose objects are sets and whose morphisms are functions.

A group can be construed as a special case of a category: it is a category with only one object, whose morphisms are all invertible. The special case of the Yoneda lemma for such a category says that the (single) object can be understood as a set $S$ and the morphisms (which are the group elements) can be understood as invertible functions from $S$ to itself.

This is exactly Cayley's theorem of group theory; it says that any group can be understood as a group of permutations of the elements of some set.

Answer 2

Some topological results generalize famously used theorems of calculus, including the intermediate value theorem and the extreme value theorem.

Recall the Intermediate Value Theorem, which states that given a continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ and values $f(a)<f(b)$ and $c$ such that $f(a)<c<f(b)$, then there exists $d\in \mathbb{R}$ such that $f(d)=c$. This is generalized by the fact that if $X$ is a connected topological space, $Y$ a topological space, and $f:X\rightarrow Y$ is a continuous function, then $f[X]$ is a connected subspace of $Y$.

Similarly, the Extreme Value Theorem states that given a continuous function $f:[a,b]\rightarrow \mathbb{R}$, then there are $c,d\in [a,b]$ such that $f(c)\leq f(x)\leq f(d)$ for every $x\in [a,b]$. This is generalized by the fact that if $X$ is a compact topological space, $Y$ a topological space, and $f:X\rightarrow Y$ a continuous function, then $f[X]$ is a compact subspace of $Y$.

Answer 3

In spherical geometry, the area of a triangle on a unit sphere with angles $\angle A, \angle B, \angle C$ is $$A + B + C - \pi,$$ a result that dates to maybe the 17th century.

The Gauss-Bonnet Theorem generalizes this to any compact surface (2-dimensional Riemannian manifold with corners) $(M, g)$: If $(M, g)$ has Gaussian curvature $K$ and its boundary $\partial M$ has geodesic curvature $k_g$, we have $$\int_M K \,ds + \int_{\partial M} k_g = 2 \pi \chi(M),$$ where $\chi(M)$ is the Euler characteristic. (Note the in interpreting the boundary integral, we must include the sum of the turning angles at each of the corners.)

I suppose this also generalizes the classic high school result that the sum of the angles of a plane triangle is $\pi$, as well as the formula for the circumference of a unit circle.

Answer 4

Lesbesgue integral is a not obviously a generalization of the Riemann integral. It is not hard to prove that every Riemann-integrable function is Lebesgue-integrable and that the values of both integrals agree whenever they are both defined.

An integral that is a generalization of Riemann integral is Henstock-Kurzweil integral. You may also check more here.

Answer 5

Law of cosines is a generalization of the Pythagorean theorem

Answer 6

Every high school student learns (or should learn) the quadratic formula $$\frac{-b \pm \sqrt{b^2 - 4 ac}}{2a},$$ which gives the roots of a polynomial $ax^2 + bx + c$.

There also turn out to be general formulas for cubic and quartic polynomials. These are not very useful practically, but their existence, together with the fact that no such formula (i.e., in terms of the usual arithmetic operations and radicals) exists for a general degree $5$ (or higher) polynomial reflects a much deeper fact, and the problem of finding a formula for quintic polynomials led to the development of Galois theory and group theory more generally. The central result, in terms of solving explicitly for roots is this:

Given any polynomial, its roots are expressible in terms of its coefficients, the usual arithmetic operations, and radicals iff its Galois group is solvable. More or less by definition, the Galois group $Gal(d)$ of a polynomial of degree $d$ is a subgroup of $S_d$; if $d \leq 4$, then $S_d$ (and hence $Gal(d)$) is solvable, reflecting the fact that explicit general formulae for the roots exist. On the other hand, $A_5 < S_5$ is simple. Some (in fact, in a certain sense, nearly all) polynomials of degree $5$ have Galois group $S_5$, so the roots of most quintics cannot be written down in the above terms.

Answer 7

The Cantor's theorem is a generalization of the elementary inequality $$n<2^n,\qquad n\in\Bbb N$$

Answer 8

The three isomorphism theorems in universal algebra that generalize the three isomorphism theorems of groups, rings, semigroups, etc.: http://en.wikipedia.org/wiki/Isomorphism_theorem#General

The special cases in groups, rings, etc. are commonly taught in upper level abstract algebra courses. But the general case is only usually found in a universal algebra course offered at graduate school level.

Answer 9

Measure theory can be thought of as a generalization of integration in $\mathbb R^n$. For example the Lebesgue measure on $\mathbb R$ extends the universe of integrable functions $\mathbb R \to \mathbb R$, including things like the characteristic of $\mathbb Q$, giving a richer theory, analogous to the theory of Riemann integration .

Answer 10

Given any function $f$ from an interval $[a, b]$ to itself, the function $g(x) := f(x) - x$ satisfies $g(a) \geq 0$ and $g(b) \leq 0$, and so by the Intermediate Value Theorem, there is at least one value $c \in [a, b]$ such that $0 = g(c) = f(c) - c$, that is at least one value $c$ that $f$ maps to itself.

This generalizes dramatically to a class of theorems commonly called the Brouwer Fixed Point Theorem. A typical example, usually called the Schauder Fixed Point Theorem says that any map from a convex, compact subset of a Banach space to itself fixes a point. Other variations very interesting in their own right include the Borsuk-Ulam Theorem and Lefschetz Fixed Point Theorem. Such theorems have found applications in differential equations and game theory among other places.

Answer 11

Two quotes from Very Basic Lie Theory

Many of the standard theorems of linear algebra are of course part of the fabric of Lie theory, and gain coherence when considered in that light. [followed by many examples]

The fact that second mixed partial derivatives are equal is a reflection of the fact that $\Bbb R^n$ is an abelian Lie group.

Answer 12

Metric spaces are a generalization of $|.|$ the absolute value.
Group are a generalization of $+$ addition and $\times$ multiplication.

Answer 13

The original Chinese remainder theorem (about integer numbers and congruences) can be generalized to the Chinese Remainder Theorem for Commutative Rings.

Answer 14

Algebraic topology is a vast generalization of Euler's formula for polyhedra: $$C+V=A+2.$$

Answer 15

Heron's formula for a triangle

$A=\sqrt{s(s-a)(s-b)(s-c)}$

generalizes to Brahmagupta's formula:

$A=\sqrt{(s-a)(s-b)(s-c)(s-d)}$

for every cyclic quadrilateral, where $A$ is the area and $a,b,c,d$ are the sides.

This generalizes even more - to any kind of convex quadrilaterals (Bretschneider's formula):

$A=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos\frac{\alpha+\beta}{2}}$

$\alpha,\beta$ are angles on opposite sides of the quadrilateral. Equivalently,

$A=\sqrt{(s-a)(s-b)(s-c)(s-d)-\frac{1}{4}(ac+bd+pq)(ac+bd-pq)}$

$p,q$ are the diagonals.

All this generalizes even more - to cyclic pentagons, hexagons, to volumes of various figures, etc.

Answer 16

The Parallelogram law of inner product spaces is a generalization of a theorem of Euclidean geometry.

Answer 17

Fubini's theorem can be interpreted as a generalization of Cavalieri's principle.

Answer 18

Cauchy-Schwarz inequality tells us that ($\forall a_i,b_i\in\mathbb R$, $i\in\{1,2,\ldots,n\}$):

$(a_1^2+a_2^2+\cdots+a_n^2)(b_1^2+b_2^2+\cdots +b_n^2)\ge (a_1b_1+a_2b_2+\cdots+a_nb_n)^2$

Whereas Hölder's inequality lets us know that ($\forall a_{ij}\in\mathbb R^{+}$, $i\in\{1,2,\ldots,m\},j\in\{1,2,\ldots,n\}$):

$(a_{11}^m+a_{12}^m+\cdots+a_{1n}^m)(a_{21}^m+a_{22}^m+\cdots+a_{2n}^m)\cdots(a_{m1}^m+a_{m2}^m+\cdots+a_{mn}^m)$

$\ge \left(a_{11}a_{21}\cdots a_{m1}+a_{12}a_{22}\cdots a_{m2}+\cdots+a_{1n}a_{2n}\cdots a_{mn}\right)^m$

The latter inequality is just a special case Hölder's inequality. The actual one states that

$\forall a_{ij}\in\mathbb R^{+}$, $i\in\{1,2,\ldots,m\},j\in\{1,2,\ldots,n\}$ and
$\forall p_i\in\mathbb R^+, i\in\{1,2,\ldots,m\}, p_1+\cdots+p_m=1$

$$(a_{11}+\cdots+a_{1n})^{p_1}\cdots(a_{m1}+\cdots+a_{mn})^{p_m}\ge a_{11}^{p_1}a_{21}^{p_2}\cdots a_{m1}^{p_m}+\cdots+a_{1n}^{p_1}a_{2n}^{p_2}\cdots a_{mn}^{p_m}$$

This one requires positive reals, so I put this requirement into the special case too. I'm not sure if the special case holds for all real numbers.

Answer 19

The BPT (Basic proportionality theorem) is a generalisation of the Mid-point theroem in geometry.

The mid point theorem states that if a line is draw joining the midpoint of two sides of a triangle then it is parellel to the base (and half of it).

The BPT states that this is true for a line which cuts the two sides in any given ratio.

I don't know if you will consider this to be "currently celebrated".

Answer 20

Lawvere's Fixed Point Theorem generalizes Cantor's Theorem, Turing's Halting Problem, Gödel's First Incompleteness Theorem, Tarski's Undefinability of Truth Theorem, the Recursion Theorem and more.

References:

Diagonal arguments and cartesian closed categories.

A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points.

Fixed points in cartesian closed categories.

Answer 21

The quartic formula is a generalization of the cubic formula, which in itself is a generalization of the quadratic formula, which in its turn is a generalization of the linear case. To verify this, take,

for instance, the quadratic formula $x_{_{1,2}}=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$, and compute its limit for $a\to0$ by

using L'Hopital. You will obtain the solution to the linear equation $bx+c=0$. In other words, the linear is a special case of the quadratic, making the latter a generalization of the former. The same goes for the other two cases as well, where the quadratic itself is a particular case of the cubic, and the cubic a corollary to the quartic, both obtained by the same method as above. $~$ Other examples would be Newton's binomial series as a generalization of the binomial theorem, and the $\Gamma$ and beta functions as generalizations of the factorial and binomial coefficients respectively. Exponentiation itself is a generalization of repeated multiplication, and even the apparently harmless operation of multiplication is nothing else than a generalization of repeated addition. If this might sound a bit odd to you, just think about how nonsensical it would be to write $e\cdot\pi$ as $\underbrace{\pi+\pi+\ldots+\pi}_{e\text{ times}}$ or even $\underbrace{e+e+\ldots+e}_{\pi\text{ times}}$. Both the Pythagorean theorem and the law of cosines have each a $3$-dimensional counterpart for pyramids. Not to mention the fact that the law of cosines is itself a two-dimensional

generalization of Pythagoras! The formula for the volume of a pyramid, $V=\dfrac{Ah}3$, extends the one

for the surface area of a triangle, $A=\dfrac{ah}2$. For n dimensions, we have $V_n=\dfrac{A_nh}n$. The same also

holds for squares, cubes, hypercubes, etc. whose perimeters, areas, volumes, etc. are given by the

general formulas $V_n=\ell^n$, and $A_n=2~n~\ell^{n-1}=2~V'_n$. For circles, spheres, hyperspheres, etc. we

have $A_n=V'_n$ , e.g., $2\pi r=\big(\pi r^2\big)'$, and $4\pi r^2=\bigg(\dfrac43\pi r^3\bigg)'$, etc. Now, I am fully aware that you probably had fancier examples in mind, but what I am trying to point out by all this is that, if not all, then at least a great part of mathematics is nothing more than a generalization or extension of previous results, with newly–added layers building upon older ones, ad infinitum. Hope this helps.