Methods of Modern Mathematical Physics, Vol 1: Functional Analysis

By C. Isham. 1972

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Equivalence of distance-based and RKHS-based statistics in hypothesis testing (2012, background): “Namely, this implies that S ˜ k P is a trace class operator, and, thus, a Hilbert– Schmidt operator [Reed and Simon (1980), Proposition VI.23].”
Wavelets on Graphs via Spectral Graph Theory (2009, positive): “One may define a measurable function of a bounded self-adjoint linear operator on a Hilbert space using the continuous functional calculus [38].”
Kernel ridge vs. principal component regression: Minimax bounds and the qualification of regularization operators (2017, methodology): “(10) Here, gλ is applied to the spectrum of the self-adjoint finite-rank operator TX via the functional calculus (see, e.g., Reed and Simon, 1980, Ch. 7); this 2This definition follows Engl et al. (1996) and Bauer et al. (2007), but is slightly more restrictive. is a generalization of standard…”
On Bayesian A- and D-optimal experimental designs in infinite dimensions (2014, background): “j=1 〈Afj , fj〉H , where the sum is finite and its value is independent of the choice of the orthonormal basis [7, 14].”
Stochastic Interface Models (2005, methodology): “To this end, one can apply Birkhoff’s individual ergodic theorem for μv , and then Stone-Weierstrass’s theorem ([218], p.”
Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe (2022, methodology): “This may be possible via the Hahn-Banach separation theorem [485, 502].”