Mathematics Summary Flashcards - MPSI/MP

Vocabulary flashcards created from the summary notes of the MPSI/MP Mathematics course covering Analysis, General Algebra, Linear Algebra, and Geometry.

Ordered Field

The set $ℑ$ of real numbers equipped with two operations $+$ and $\times$ compatible with the order relation $\leqslant$, satisfying $\forall a, b, c \in \mathbb{R}, a \leqslant b \implies a + c \leqslant b + c$ and $\forall a, b \in \mathbb{R}, \forall c \geqslant 0, a \leqslant b \implies ac \leqslant bc$.

Archimedean Property

The property of real numbers stating that for any $a \in \mathbb{R}$ and $b > 0$, there exists an integer $k \in \mathbb{N}$ such that $bk > a$.

Neighborhood of a Point

A subset $V$ of $\mathbb{R}$ that contains an open interval centered at $a \in \mathbb{R}$.

Supremum (Borne sup erieure)

The smallest element (if it exists) of the set of upper bounds (majorants) of a subset $A \subset \mathbb{R}$.

Floor Function (Partie entiere)

For a real number $x$, the unique largest relative integer denoted $E(x)$ or $[x]$ satisfying $E(x) \leqslant x < E(x) + 1$.

Lipschitz Function

A function $f$ defined on $D$ such that there exists $k \geqslant 0$ satisfying $\forall x, y \in D, |f(x) - f(y)| \leqslant k |x - y|$. It is called contracting when $k < 1$.

Uniform Continuity

A property of a function $f$ on $D$ defined by $\forall \varepsilon > 0, \exists \alpha > 0, \forall x, x' \in D, |x - x'| \leqslant \alpha \implies |f(x) - f(x')| \leqslant \varepsilon$, where $\alpha$ depends on $\varepsilon$ but not on $x$.

Heine's Theorem

The theorem stating that every continuous function on a closed bounded interval (segment) is uniformly continuous on that segment.

Rolle's Theorem

The theorem stating that if a function $f$ is continuous on $[a,b]$, differentiable on $]a,b[$, and satisfies $f(a) = f(b)$, then there exists at least one point $c \in ]a,b[$ such that $f'(c) = 0$.

Convex Function

A function $f$ defined on an interval $I$ such that any arc of its representative curve lies below the corresponding chord, satisfying $\forall x_1, x_2 \in I, \forall k \in [0,1], f(k x_1 + (1-k)x_2) \leqslant k f(x_1) + (1-k)f(x_2)$.

Piecewise Continuous Function

A function $f$ defined on $[a,b]$ for which there exists a subdivision $(x_i)_{0 \leqslant i \leqslant n}$ such that $f$ is continuous on each open interval $]x_i, x_{i+1}[$ and admits finite left and right limits at every subdivision point.

Norm

A map $N$ from a vector space $E$ into $\mathbb{R}$ satisfying positivity and definiteness ($N(x) \geqslant 0$ and $N(x) = 0 \iff x = 0$), absolute homogeneity ($N(\lambda x) = |\lambda| N(x)$), and the triangle inequality ($N(x+y) \leqslant N(x) + N(y)$).

Open Set

A subset $A$ of a normed vector space $E$ that is a neighborhood of each of its points, satisfying $\forall a \in A, \exists r_a > 0, B(a, r_a) \subset A$.

Compact Set

In a normed space $E$, a subset $A$ such that from every sequence of elements of $A$, one can extract a subsequence that converges to a limit in $A$. In finite dimension, it is equivalent to being closed and bounded.

Banach Space

A normed vector space $E$ that is complete, meaning that every Cauchy sequence in $E$ converges to an element of $E$.

Schwarz's Theorem

The theorem stating that if at least one of the second partial derivatives $\frac{\partial^2 f}{\partial x_i \partial x_j}$ or $\frac{\partial^2 f}{\partial x_j \partial x_i}$ of a function $f$ is continuous at point $a$, then $\frac{\partial^2 f}{\partial x_i \partial x_j}(a) = \frac{\partial^2 f}{\partial x_j \partial x_i}(a)$.

Jacobian Matrix

For a function $f = (f_1, \dots, f_p): D \subset \mathbb{R}^n \to \mathbb{R}^p$ of class $C^1$ at $a$, the $p \times n$ matrix $J_f(a)$ whose $(i,j)$-th entry is the partial derivative $\frac{\partial f_i}{\partial x_j}(a)$.

Exact Differential Form

A differential form $\omega$ of degree $1$ on an open set $U$ for which there exists a scalar function $f: U \to \mathbb{R}$ such that $df = \omega$.

Normal Convergence

A mode of convergence for a function series $\sum u_n$ on an interval $I$ defined by the convergence of the numerical series of supremum norms $\sum \|u_n\|_\infty$.

Radius of Convergence

For a power series $\sum a_n z^n$, the number $R \in [0, +\infty]$ such that the series converges absolutely for all $|z| < R$ and diverges for all $|z| > R$.

Dirichlet's Theorem

The theorem stating that if $f$ is $T$-periodic and piecewise $C^1$ on a segment of length $T$, its Fourier series converges at every point $t \in \mathbb{R}$ to its regularized value $\frac{f(t^+) + f(t^-)}{2}$.

Injective Application

An application $f: E \to F$ satisfying $\forall x, x' \in E, f(x) = f(x') \implies x = x'$.

Group

A non-empty set $G$ equipped with an internal binary operation $\ast$ that is associative, has an identity element $e$, and for which every element in $G$ possesses a symmetric (inverse) element.

Ring

A set $A$ equipped with two internal operations $+$ and $\times$ such that $(A, +)$ is a commutative group, $\times$ is associative, has an identity element, and is distributive over $+$.

Field

A non-zero ring in which every non-zero element has a multiplicative inverse.

d'Alembert-Gauss Theorem

The fundamental theorem of algebra stating that every non-constant polynomial with complex coefficients has at least one root in $\mathbb{C}$.

Rank-Nullity Theorem

For a linear application $f$ defined on a finite-dimensional vector space $E$, the identity $\text{dim}(E) = \text{dim}(\text{Ker}(f)) + \text{rg}(f)$, where $\text{rg}(f) = \text{dim}(\text{Im}(f))$.

Dual Space

The vector space $L(E, K)$ of all linear forms on a vector space $E$, denoted $E^*$. If $E$ is finite-dimensional, $\text{dim}(E^*) = \text{dim}(E)$.

Eigenvalue and Eigenvector

For an endomorphism $f \in L(E)$, a non-zero vector $x \in E$ is an eigenvector if there exists a scalar $\lambda \in K$ such that $f(x) = \lambda x$; the scalar $\lambda$ is the corresponding eigenvalue.

Cayley-Hamilton Theorem

The theorem stating that for any endomorphism $f$ on a finite-dimensional vector space, its characteristic polynomial $P_f$ is an annihilating polynomial for $f$, meaning $P_f(f) = 0$.

Euclidean Vector Space

A finite-dimensional real vector space equipped with a real inner product (a positive-definite, symmetric bilinear form).

Orthogonal Endomorphism

An endomorphism $f$ on a Euclidean vector space $E$ that preserves the inner product, satisfying $\forall x, y \in E, \langle f(x) | f(y) \rangle = \langle x | y \rangle$.

Symmetric Endomorphism

An endomorphism $f$ on a Euclidean vector space $E$ that equals its adjoint $f^*$, satisfying $\forall x, y \in E, \langle f(x) | y \rangle = \langle x | f(y) \rangle$.