In the previous essays, we discussed operators and the hierarchy of Topological spaces on which they may act. Now we will begin connecting fundamental concepts from linear algebra to operator theory to build deeper intuition. We will derive the Rank-Nullity Theorem and show how it connects naturally to the Fredholm index. Assuming we have a … Continue reading On the Fredholm Index (Essay III)
Course Introduction This course is based primarily on John B. Conway's "A Course in Operator Theory" book. The goal is to develop a rigorous foundation suitable for graduate-level pure mathematics, while presenting the material in a way that remains meaningful to students in the applied sciences. Operator Theory sits naturally at the intersection of analysis, … Continue reading Introduction to Operator Theory (Essay I)
As I develop the BGecko course Lectures on Operator Theory and Operator Algebras, it is essential to establish a rigorous overview on the hierarchy of topological spaces. In our previous essay, we defined the operator, and how they might be interpreted over both finite and infinite dimensional spaces. In Linear Algebra, we study the geometry … Continue reading On the Hierarchy of Topological Spaces (Essay II)
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"The art of doing mathematics consists in finding that special case which contains all germs of generality." -- David Hilbert. The Gamma function appears in many areas of mathematics, physics, and engineering. This post will not only provide motivation for studying the Gamma function but also present several methods for deriving it, including an introduction … Continue reading Developmental – Welcoming The Gamma Function
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