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A short remark concerning pedagogy and group homomorphisms
Imagine you are sitting in your first course in abstract algebra. In the last few classes, your instructor has introduced the notion of a group, proved some basic propositions about them, and motivated the group concept with a variety of examples. “Today’s topic,” your instructor announces, “is group homomorphisms. Before resorting to a formal definition, […]
The Riesz representation theorem: Part III
Part III: Uniqueness in the compact Hausdorff case The goal of this post is to shed light on the uniqueness part of the Riesz representation theorem for positive linear functionals on when is a compact Hausdorff space. III.1 Definition: Let be a compact Hausdorff space. A measure on whose -algebra of definition contains the Borels […]
The Riesz representation theorem: Part II
Part II: “Why ?” for operator algebraists Let be a locally compact Hausdorff space. To anybody interested in C*-algebras, the study of the space of continuous functions which vanish at infinity needs no motivation. Gelfand theory shows that the space and the C*-algebra are interchangeable. Any statement that can be made about can be rephrased […]
The Riesz representation theorem: Part I
This week I’ve decided to firm up my understanding of the Riesz representation theorem. Since there are many “Riesz representation theorems”, I should clarify that I am referring to the Riesz-Markov-Kakutani representation theorem which realizes each positive linear function on , a locally compact Hausdorff space, as integration against a measure. The main source I’m […]
Unneccessary And Insufficient