One of the pillars of functional analysis is the Hahn-Banach theorem, so it makes sense to dedicate a post to this theorem. On normed spaces, the theorem has a plethora of interesting corollaries, some of which will be stated here. The locally convex spaces are of interest since they are the most rudimentary topological vector spaces on which the the Hahn-Banach theorem can be used to extend continuous linear functionals, and encompasses a sizable chunk of the topological vector spaces one might meet in the wild.
Topological Complements
– Introduction –
The first steps outside the comforts of the category of Hilbert spaces, the safe space for of functional analysis, into the unruly world of topological vector spaces, can be a troubling experience for any student, myself included. To easy the passage, here are a few tips and results regarding the existence of complementary subspaces in the general setting of topological vector spaces. For Hilbert spaces it is known that every closed subspace has a preferred (topologically) complementary subspace, namely the orthogonal complement, but any two (algebraically) complementary closed subspaces are automatically (topologically) complementary (by Theorem 1).
Continue reading “Topological Complements”
Three different entropies, variational principle and the degree formula.
My inaugural blog post, hurray!! This post is based on a mandatory assignment in a course in complex dynamics held at the university of Oslo 2018. For sources we mainly the book of Walters [1] and handouts. Some of the more basic results will, due to time constraints, be left unproven, but can all be found in [1].
– Introduction –
The notion of entropy, first formulated in the context of thermodynamics, is a quantity meant to capture the complexity/chaoticness of the state of a system. It is maybe not surprising that such a broadly defined notion spread like wild fire throughout different subfields of applied mathematics and information theory. Here, three different definitions of entropy are introduced, two of which turn out to be equivalent when working on compact metric spaces, and all three are related, on compact metric spaces, by the so called Variational Principle.
Lastly, the theorem of Misiurewicz, Przytycki and Gromov, which relates the entropy of a holomorphic map on the
-sphere to its degree was proven.
Lastly, the theorem of Misiurewicz, Przytycki and Gromov, which relates the entropy of a holomorphic map on the
-sphere to its degree was proven.Continue reading “Three different entropies, variational principle and the degree formula.”