One of the central facts that turns functional analysis into a non-trivial generalization of linear algebra is the lack of compactness phenomena that arises in the infinite dimensional analysis. Indeed, in virtue of Riesz's lemma the closed balls are never sequentially compact (and therefore never compact) in infinite dimensional NLS's. On the other hand, since … Continue reading Variational methods and Weak Topologies: Part 1
So far, the examples of (Fréchet) differentiable functions presented are all classical in some sense and even though there was necessary to introduce some new lemmas to compute explicitly the derivative, the computations have been reasonable until now. However, in many practical examples it is not possible to compute the Fréchet derivative in one step. … Continue reading Calculus in Banach spaces: Gateaux derivative and consecuences of the mean value theorem
The first two examples presented in the begining of these notes show us that the space of modelling of a PDE is quite related to its formulation. So far we have presented two types of problems: the classical Dirichlet problem and the weak variational problem. The former involves a pointwise statement and the later an … Continue reading Calculus in Banach Spaces: Application to classical functionals
The study of linear equations, known as linear algebra in the finite dimensional case and as functional analysis in the general case, has furnished powerful and beautiful techniques to prove well-posedness of a huge and diverse amount of problems, like the weak variational problem commented in the last part. Many results like Lax-Milgram theorem, Fredholm … Continue reading Calculus in Banach Spaces: Fréchet Derivative.