Undergraduate mathematics in mathlib

This gives pointers to undergraduate maths topics that are currently covered in mathlib. There is also a page listing undergraduate maths topics that are not yet in mathlib.

Linear algebra

Fundamentals vector space, product of vector spaces, vector subspace, quotient space, sum of subspaces, complementary subspaces, linear independence, generating sets, bases, existence of bases, linear map, range of a linear map, kernel of a linear map, algebra of endomorphisms of a vector space, general linear group.

Duality dual vector space, dual basis, transpose of a linear map.

Finite-dimensional vector spaces finite-dimensionality, isomorphism with $K^n$, rank of a linear map, isomorphism with bidual.

Multilinearity multilinear map, determinant of vectors.

Matrices commutative-ring-valued matrices, field-valued matrices, matrix representation of a linear map, determinant, invertibility.

Endomorphism polynomials minimal polynomial, characteristic polynomial, Cayley-Hamilton theorem.

Structure theory of endomorphisms eigenvalue, eigenvector, generalized eigenspaces.

Group Theory

Basic definitions group, group morphism, direct product of groups, subgroup, subgroup generated by a subset, order of an element, normal subgroup, quotient group, group action, stabilizer of a point, orbit, quotient space, conjugacy classes.

Abelian group cyclic group, complex roots of unity, primitive complex roots of unity.

Permutation group permutation group of a type, decomposition into transpositions, decomposition into cycles with disjoint support, signature, alternating group.

Classical automorphism groups general linear group, special linear group, orthogonal group, unitary group.

Ring Theory

Fundamentals ring, subrings, ring morphisms, ring structure $\Z$, product of rings.

Ideals and Quotients ideal of a commutative ring, quotient rings, prime ideals, maximal ideals, Chinese remainder theorem.

Algebra associative algebra over a commutative ring.

Divisibility in integral domains irreducible elements, invertible elements, coprime elements, unique factorisation domain (UFD), greatest common divisor, least common multiple, $A[X]$ is a UFD when $A$ is a UFD, principal ideal domain, Euclidean rings, Euclid's' algorithm, $\Z$ is a euclidean ring, congruence in $\Z$, prime numbers, $\Z/n\Z$ and its invertible elements, Euler's totient function ($\varphi$).

Polynomial rings $K[X]$ is a euclidean ring when $K$ is a field, irreducible polynomial, cyclotomic polynomials in $\Q[X]$, Eisenstein's criterion, polynomial algebra in one or several indeterminates over a commutative ring, roots of a polynomial, multiplicity, polynomial derivative, decomposition into sums of homogeneous polynomials, symmetric polynomials.

Field Theory fields, characteristic of a ring, characteristic zero, characteristic p, Subfields, Frobenius morphisms, field $\Q$ of rational numbers, field $\R$ of real numbers, field $\C$ of complex numbers, $\C$ is algebraically closed, field of fractions of an integral domain, algebraic elements, algebraic extensions, algebraically closed fields, rupture fields, splitting fields, finite fields.

Bilinear and Quadratic Forms Over a Vector Space

Bilinear forms bilinear forms, alternating bilinear forms, symmetric bilinear forms, nondegenerate forms, matrix representation, change of coordinates.

Quadratic forms quadratic form, polar form of a quadratic.

Orthogonality orthogonal elements.

Euclidean and Hermitian spaces Euclidean vector spaces, Hermitian vector spaces, dual isomorphism in the euclidean case, orthogonal complement, Cauchy-Schwarz inequality, norm, orthonormal bases.

Endomorphisms self-adjoint endomorphism.

Affine and Euclidean Geometry

General definitions affine space, affine function, affine subspace, barycenter, affine span.

Euclidean affine spaces angles between vectors.

Single Variable Real Analysis

Real numbers definition of $\R$, field structure, order.

Sequences of real numbers convergence, limit point, recurrent sequences, limit infimum and supremum, Cauchy sequences.

Topology of R metric structure, completeness of R, Bolzano-Weierstrass theorem, compact subsets of $\R$, connected subsets of $\R$, additive subgroups of $\R$.

Numerical series Geometric series, convergence of $p$-series for $p>1$.

Real-valued functions defined on a subset of $\R$ continuity, limits, intermediate value theorem, image of a segment, continuity of monotonic functions, continuity of inverse functions.

Differentiability derivative at a point, differentiable functions, derivative of a composition of functions, derivative of the inverse of a function, Rolle's theorem, mean value theorem, higher order derivatives of functions, $C^k$ functions, Leibniz formula.

Elementary functions (trigonometric, rational, $\exp$, $\log$, etc) polynomial functions, logarithms, exponential, power functions, trigonometric functions, hyperbolic trigonometric functions, inverse trigonometric functions, inverse hyperbolic trigonometric functions.

Integration antiderivative of a continuous function, integration by parts.

Sequences and series of functions uniform convergence, continuity of the limit.

Convexity convex functions of a real variable, characterizations of convexity, convexity inequalities.

Single Variable Complex Analysis

Complex Valued series radius of convergence, continuity, differentiability with respect to the complex variable, complex exponential, extension of trigonometric functions to the complex plane.


Topology and Metric Spaces topology of a metric space, induced topology, finite product of metric spaces, limits of sequences, cluster points, continuous functions, homeomorphisms, compactness in terms of open covers (Borel-Lebesgue), sequential compactness is equivalent to compactness (Bolzano-Weierstrass), connectedness, connected components, path connectedness, Lipschitz functions, uniformly continuous functions, Heine-Cantor theorem, complete metric spaces, contraction mapping theorem.

Normed vector spaces on $\R$ and $\C$ topology on a normed vector space, Banach open mapping theorem, equivalence of norms in finite dimension, norms $\lVert\cdot\rVert_p$ on $\R^n$ and $\C^n$, absolutely convergent series in Banach spaces, continuous linear maps, norm of a continuous linear map, uniform convergence norm (sup-norm), normed space of bounded continuous normed-space-valued functions, its completeness, Heine-Borel theorem (closed bounded subsets are compact in finite dimension), Arzela-Ascoli theorem.

Hilbert Spaces Hilbert projection theorem, orthogonal projection onto closed vector subspaces, dual space, Riesz representation theorem, inner product spaces $l^2$ and $L^2$, their completeness, Hilbert (orthonormal) bases (in the separable case).

Multivariable calculus

Differential Calculus differentiable functions on an open subset of $\R^n$, differentials (linear tangent functions), chain rule, mean value theorem, differentiable functions, $k$-times continuously differentiable functions, local extrema, convexity of functions on an open convex subset of $\R^n$, diffeomorphisms, inverse function theorem, implicit function theorem.

Differential equations Grönwall lemma.

Measures and integral Calculus

Measure theory measurable spaces, sigma-algebras, product of sigma-algebras, Borel sigma-algebras, positive measure, counting measure, Lebesgue measure, product measure, measurable functions, approximation by step functions.

Integration integral of positive measurable functions, monotone convergence theorem, Fatou's lemma, integrable functions, dominated convergence theorem, finite dimensional vector-valued integrable functions, $\mathrm{L}^p$ spaces where $1 ≤ p ≤ ∞$, Completeness of $\mathrm{L}^p$ spaces, Holder's inequality, Fubini's theorem.

Probability Theory

Definitions of a probability space probability measure, independent events, sigma-algebra, independent sigma-algebra, Borel-Cantelli lemma (easy direction).

Random variables and their laws discrete law, independence of random variables, Bernoulli law.

Convergence of series of random variables almost surely convergence.