MA3G7 Functional Analysis I Notes | Assignment Help | Syllabus
Country: United Kingdom
University: The University Of Warwick
This is a module on infinite-dimensional Hilbert spaces, which arise frequently in practical mathematics. The ideas presented here allow for a comprehensive understanding of Fourier series and the theory of Sturm-Liouville boundary value problems in general. They’re also at the heart of modern partial differential equations theory.
Despite the fact that Hilbert spaces need an infinite number of basis elements, they retain many of the known properties of finite-dimensional Euclidean spaces ( mathbbRn ), such as the inner product and related notions of length and distance. The unlimited dimensionality of the spaces offers up new possibilities, and much of the theory is devoted to reclaiming control over them given the proper conditions.
The module is broken down into three parts. In the first section, we’ll look at several famous Hilbert spaces and examine how they relate geometrically to Euclidean spaces (inner product, expansion in terms of basis elements, etc.). Then we’ll examine at a few other definitions of convergence in a Hilbert space, which differ in this instance while being equal in finite-dimensional spaces. Finally, we examine characteristics of linear operators between Hilbert spaces (equivalent to the theory of matrices between finite-dimensional spaces), finding conclusions for a specific class of such operators that are remarkably similar to those found in the finite-dimensional scenario (compact self-adjoint operators).
To validate and illustrate the abstract theory, more concrete instances will be utilized.
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