A common theme in mathematical analysis (particularly in analysis of a “geometric” or “statistical” flavour) is the interplay between “macroscopic” and “microscopic” scales. These terms are somewhat vague and imprecise, and their interpretation depends on the context and also on one’s choice of normalisations, but if one uses a “macroscopic” normalisation, “macroscopic” scales correspond to scales that are comparable to unit size (i.e. 6,993 more words

## Tags » Gleason-Yamabe Theorem

#### 254A, Notes 5: The structure of locally compact groups, and Hilbert's fifth problem

In the previous notes, we established the *Gleason-Yamabe theorem*:

4,112 more wordsTheorem 1 (Gleason-Yamabe theorem) Let be a locally compact group. Then, for any open neighbourhood of the identity, there exists an open subgroup of and a compact normal subgroup of in such that is isomorphic to a Lie group.

#### 254A, Notes 4: Building metrics on groups, and the Gleason-Yamabe theorem

In this set of notes we will be able to finally prove the Gleason-Yamabe theorem from Notes 0, which we restate here:

6,874 more wordsTheorem 1 (Gleason-Yamabe theorem) …

#### 254A, Notes 3: Haar measure and the Peter-Weyl theorem

In the last few notes, we have been steadily reducing the amount of regularity needed on a topological group in order to be able to show that it is in fact a Lie group, in the spirit of… 6,977 more words