Two Deletion Theorems

How many of us would bet that in a normed space the complementary of a one-point set could be homeomorphic to the entire space? And yet… In an infinite-dimensional space it is always true!
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Abstract Semisimplicity

In linear algebra semi-simplicity can be found in various contexts. They all derive from the theory of semisimple modules, which can be seen as a generalization of elementary results concerning direct sums of linear spaces. But when looking more closely we see that the whole theory does not even make use of the module structure: it only uses the properties of the lattice of submodules. Notions and results concerning semisimplicity (direct sums, semisimplicity, isotypical components, multiplicity) all survive in an abstract framework where submodules are replaced by points of a lattice provided with the appropriate axioms.
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Stone's Representation Theorem

The following result is well-known and is usually called Stone's representation theorem:

Every Boolean algebra is isomorphic to that of simultaneously open and closed sets of some suitable topological space. Moreover the latter can be chosen to be compact Haussdorf and totally discontinuous, and in this case is unique up to homeomorphism.

However Stone's statement is more general. It deals with any distributive lattice and illustrates the truly innovative ideas that we owe to Stone. Among them one can cite the fundamental idea of considering (before Zariski) prime ideals as points, as well as a genuine result of duality several years before the formulation of categorical equivalence by Eilenberg and Mac Lane.
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A Characterization of Moments of Inertia of a Solid Body

In an affine Euclidean space \(\E\) of dimension \(n≥3,\) let us consider a solid body \(\S.\) It can be given by the data consisting of a real measure \(\rho\) on the Lebesgue \(\sigma\)-algebra of \(\E,\) with compact support to avoid convergence problemswithout much interest here; we are mainly concerned in the algebraic aspects.. We can consider, for each affine subspace \(\F\) of \(\E,\) the moment of inertia \(I_{\S}(\F)\) of \(\S\) with respect to the affine subspace \(\F.\) The map \(f:\F\mapsto I_{\S}(\F)\) satisfies the following three conditions:

1. Perpendicular additivity ;
2. Huygens' theorem ;
3. Continuity for the Grassmannian topology.

Conversely, let's now forget the solid body and keep only its inertial trace on the affine subspaces: consider a map \(f\) defined on the set of affine subspaces of \(\E\) satisfying the three conditions above. We show that there then exists a solid body \(\S\) whose moments of inertia are represented by \(f\). This result constitutes in some way the affine version of Gleason's theoremAndrew M. Gleason, Measures on the closed subspaces of a Hilbert space. J. math. Mech. 6. (1957), 885-893.. We also show that condition (ii) is useless, in other words: perpendicular additivity and continuity imply Huygens' theorem.
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August 01, 2023 : Birth of this Blog

Here we go, I decided to switch from my old site to a real blog. I will now have the bare minimum to write mathematics online and spend my upcoming retirement usefully.
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