From Hilbert Spaces to Animated Elliptic PDEs: My GitHub Gist on Numerical Methods & Functional Analysis
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This content provides a comprehensive overview of Hilbert Spaces and Operators within Functional Analysis. It delves into the geometric properties of Hilbert spaces, explaining concepts like inner product spaces, orthonormal bases, and completeness. Key theorems such as the Gram-Schmidt orthogonalization, Parallelogram identity, and Riesz representative theorem are also highlighted.
The text then transitions to Operators, discussing various types including orthogonal projections, linear and bilinear forms, continuous and compact operators, and the spectral theorem. It also covers different notions of convergence, emphasizing weak convergence, and introduces the generalized eigenvalue problem. The overall aim is to establish a strong foundation for understanding these fundamental concepts in Functional Analysis and their broad applications.
