We give some background information about the Cauchy-Schwarz inequality including its history. We then continue by providing a number of proofs for the inequality in its classical form using various proof techniques, including proofs without words. Next we build up the theory of inner product spaces from metric and normed spaces and show applications of the Cauchy-Schwarz inequality in each content, including the triangle inequality, Minkowski's inequality and Hölder's inequality. In the final part we present a few problems with solutions, some proved by the author and some by others.

Au cours de ma thèse, je me suis intéressé à deux aspects de l'interaction charge-rayonnement dans des jonctions tunnel. Premièrement j'ai étudié la dynamique des transferts de charge à travers une jonction tunnel normale lorsque celle-ci est couplée à un environnement dont l'impédance est comparable au quantum d'impédance (RK= h/e^2). Nous avons montré que les uctuations de courant portent des signatures de processus à un, deux et trois photons. Deuxièmement j'ai caractérisé le rayonnement émis par une jonction Josephson lorsque celle-ci est couplée à deux résonateurs de fréquences différentes, nu_1 et nu_2, et soumise à une tension 2eV = h nu_1+h nu_2. Nous avons montré que les photons sont émis pas paires, ce qui constitue un rayonnement non-classique violant l'inégalité de Cauchy-Schwarz.Nos résultats montrent que le blocage de Coulomb dynamique constitue une ressource pour la production de micro-ondes quantiques. / In this thesis work, I have focused on two aspects of the charge-light interaction for tunnel junctions. On one hand, I have investigated the influence of the coupling to electromagnetic radiation on the dynamics of charge transfer through a normal tunnel junction when the impedance of its environment gets comparable to the resistance quantum (RK=h/e^2). We showed that the current fluctuations bear signature of the processes where one, two or three photons are emitted.On the other hand, we used a Josephson junction, embedded in a circuit made of two resonators at different frequencies, nu_1 and nu_2, and biased at the voltage 2eV=h nu_1+h nu_2. We showed that the Josephson junction emits photon pairs corresponding to a non classical state of radiation which violates the Cauchy-Schwarz inequality.Our results extend the toolbox for performing quantum optics experiments in the microwave domain.