Author: Dr. Rakesh Kumar Parmar
DOI Link: https://doi.org/10.70798/Bijmrd/04011004
Abstract: Functional analysis gives a strong and clear mathematical base for understanding artificial intelligence (AI). Today, most AI systems work in very large or even infinite-dimensional spaces, and for this we use ideas from Banach spaces, Hilbert spaces, operator theory, spectral theory, and approximation theory. These ideas help us understand how AI models represent information, how they learn, and how they perform on new data. In this paper, we look at artificial intelligence from the viewpoint of functional analysis. We explain both the basic theory and practical use. We show how concepts of functional analysis shape the structure of learning models, control the power of neural networks, guide the behavior of kernel methods, and help in making optimization stable. We also discuss recent methods in deep learning such as transformers, diffusion models, and very wide neural networks, and explain how they can be understood as operators working on function spaces. In the end, we conclude that functional analysis is not just an extra mathematical tool; it is a basic foundation that connects and explains many different techniques used in modern AI.
Keywords: Functional Analysis, Artificial Intelligence, Neural Operator, Machine Learning, Optimization, Spectral Properties.
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