space
52 theorems carry this thread across the degree.
The algebra of limits
Limits respect sums, products, and quotients.
The squeeze theorem
A sequence trapped between two convergents to L also converges to L.
The monotone convergence theorem
A bounded monotone sequence converges.
The Bolzano–Weierstrass theorem
Every bounded sequence has a convergent subsequence.
Completeness of the reals
A sequence of reals converges if and only if it is Cauchy.
The ratio test
A series converges absolutely when the limiting ratio of terms is below one.
The alternating series test
An alternating series with terms decreasing to zero converges.
The intermediate value theorem
A continuous function takes every value between two of its values.
The Heine–Borel theorem
A subset of R^n is compact iff it is closed and bounded.
Uniform continuity on compact sets
Continuous on a compact set implies uniformly continuous.
The Riemann integrability criterion
A bounded function is integrable iff upper and lower sums can be made arbitrarily close.
Uniform limits of continuous functions
A uniform limit of continuous functions is continuous.
The Weierstrass M-test
A dominated series of functions converges uniformly.
Change of variables and the Jacobian
How volumes transform under a smooth change of coordinates.
The method of Lagrange multipliers
Constrained extrema occur where gradients align.
Green's theorem
A planar circulation integral equals a double integral of curl.
Stokes' theorem
Circulation around a boundary equals the flux of curl through the surface.
The divergence theorem
Flux through a closed surface equals the integral of divergence inside.
The Cauchy–Riemann equations
Complex differentiability is equivalent to a pair of PDEs.
Cauchy's integral theorem
The integral of a holomorphic function round a closed loop is zero.
Cauchy's integral formula
A holomorphic function's values are determined by its boundary values.
Liouville's theorem
A bounded entire function is constant.
The fundamental theorem of algebra
Every non-constant polynomial over C has a root.
The residue theorem
A contour integral equals the sum of enclosed residues.
Continuity via preimages
A map is continuous iff preimages of open sets are open.
Characterisations of compactness
Open-cover and sequential compactness coincide in metric spaces.
The Banach fixed-point theorem
A contraction on a complete space has a unique fixed point.
The Baire category theorem
A complete metric space is not a countable union of nowhere-dense sets.
Tychonoff's theorem
An arbitrary product of compact spaces is compact.
Carathéodory's extension theorem
A premeasure extends to a genuine measure.
The monotone convergence theorem
Integrals commute with increasing limits of non-negative functions.
Fatou's lemma
The integral of a liminf is at most the liminf of integrals.
The dominated convergence theorem
A dominated pointwise limit may be integrated term by term.
The Fubini–Tonelli theorem
When iterated integrals may be exchanged.
The Hahn–Banach theorem
Bounded functionals extend without increasing norm.
The open mapping theorem
A surjective bounded operator between Banach spaces is open.
The closed graph theorem
A closed-graph operator between Banach spaces is bounded.
The uniform boundedness principle
Pointwise-bounded families of operators are uniformly bounded.
The Riesz representation theorem
Every bounded functional on a Hilbert space is an inner product.
Separation of variables
Reducing a PDE to ODEs via product solutions.
d'Alembert's solution
The 1D wave equation solved by travelling waves.
The maximum principle
A harmonic function attains its extrema on the boundary.
Convergence of Fourier series
When and how a periodic function equals its Fourier series.
The method of characteristics
First-order PDEs solved along characteristic curves.
The Frenet–Serret formulas
Curvature and torsion determine a space curve.
The first fundamental form
Lengths and angles on a surface from its metric.
Gauss's Theorema Egregium
Gaussian curvature is intrinsic to the surface.
The Gauss–Bonnet theorem
Total curvature is a topological invariant.
The fundamental group
Loops up to homotopy form a group invariant of the space.
The Seifert–van Kampen theorem
The fundamental group of a union from those of its pieces.
The Brouwer fixed-point theorem
Every continuous self-map of a disc has a fixed point.
Invariance of the Euler characteristic
V−E+F is a topological invariant.