Diametral dimensions and some applications to spaces Snu

The "classic" diametral dimension is a topological invariant which characterizes Schwartz and nuclear locally convex spaces. Besides, there exists a second diametral dimension which is conjectured to be equal to the first one (on Fréchet-Schwartz spaces). The first part of this thesis is dedicated to the study of this conjecture. We present several positive partial results in metrizable spaces (in particular in Köthe sequence spaces and Hilbertizable spaces) and some properties which provide the equality of the two diametral dimensions (such as the Delta-stability, the existence of prominent bounded sets, and the property Omega bar). Then, we describe the construction of some non-metrizable locally convex spaces for which the two diametral dimensions are different. The other purpose of this work is to pursue the topological study of sequence spaces Snu, originally defined in the context of multifractal analysis. For this, the second part of the present thesis focuses on the study of the two diametral dimensions in spaces Snu. Finally, we show that some classes of spaces Snu verify (a variation of) the property Omega bar