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Applications of Functional Analysis Quantum Mechanics: Functional analysis provides the mathematical foundation for quantum theory, particularly through Hilbert spaces used to describe quantum states. Differential & Integral Equations: It plays a critical role in formulating and solving various types of equations encountered in physics and engineering. Signal Processing: Concepts like Fourier analysis and function spaces help in analyzing and filtering signals across multiple domains. Numerical Analysis: It aids in developing algorithms for approximating solutions to complex mathematical problems.
Key Topics in Functional Analysis Hahn–Banach Theorem Normed Vector Spaces Open Mapping Theorem Banach Spaces Closed Graph Theorem Inner Product Spaces Uniform Boundedness Principle Hilbert Spaces Compact Operators Linear Operators and Functionals Spectral Theory Bounded and Unbounded Operators Topological Vector Spaces Dual Spaces Operator Norms
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