The definition of compact subsets of a metric space is as follows: Définition: Soit \(E\) be a metric space. A subset \(A\subset E\) is compact from any open covering of \(A\) we can find a finite subcovering , i.e. …

Let us switch to physics for a while with this link to the complete physics courses given by Feynman between 61 and 64 at Caltech. It has historical value, with original audio recordings and photos of the period, but above all pedagogical value, with all a…

In this exercise on Euclidean norms, we needed to know that the set of matrices with determinant \(>0\) is connected. Here's a reminder of the proof: Lemma: Let \(GL_n^+(\mathbb{R})=\{M\in M_n(\mathbb{R}),\det M >0\}\) and \(GL_n^-(\mathbb{R})=\{M\in M_…

An analysis-algebra problem that starts very quietly with the first 8 questions, then goes on with sequences of functions with values in \(R_n[X]\) whose coefficients are sums of integer series. Lots of algebra in the final section. The main difficulty of …

A difficult, demanding and very long algebra problem, which studies the quaternion field : matrix representation, descriptions of direct orthogonal endomorphisms, automorphisms. Many topics that are less frequently covered in other competitive exams: 1st y…