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This is a tutorial and survey paper on kernels, kernel methods, and relatedfields. We start with reviewing the history of kernels in functional analysisand machine learning. Then, Mercer kernel, Hilbert and Banach spaces,Reproducing Kernel Hilbert Space (RKHS), Mercer's theorem and its proof,frequently used kernels, kernel construction from distance metric, importantclasses of kernels (including bounded, integrally positive definite, universal,stationary, and characteristic kernels), kernel centering and normalization,and eigenfunctions are explained in detail. Then, we introduce types of use ofkernels in machine learning including kernel methods (such as kernel supportvector machines), kernel learning by semi-definite programming, Hilbert-Schmidtindependence criterion, maximum mean discrepancy, kernel mean embedding, andkernel dimensionality reduction. We also cover rank and factorization of kernelmatrix as well as the approximation of eigenfunctions and kernels using theNystr{\"o}m method. This paper can be useful for various fields of scienceincluding machine learning, dimensionality reduction, functional analysis inmathematics, and mathematical physics in quantum mechanics.

Reproducing Kernel Hilbert Space, Mercer's Theorem, Eigenfunctions, Nystr\"om Method, and Use of Kernels in Machine Learning: Tutorial and Survey