Modern analysis is more than Riemann Integration (which is more akin to "classical" analysis).
Modern analysis includes Lebesgue Integration (far more widely applicable than Riemann) and the notion of Hilbert Spaces (henceforth abbreviated to H-Spaces).
H-Spaces are a generalisation of the notion of Euclidean space and used significantly in partial differential equations and quantum mechanics; the Sturm-Liouville theory, for example, which studies a particular second-order differential equation, utilises Hilbert spaces intrinsically.
H-Spaces have the very pleasant property that they are complete - any sequence of points in the space are guaranteed to converge to a point that actually lives within the space.
Spectral theory is another "advanced" analysis concept introduced by Mr Hilbert. It extends the eigenvalue-eigenvector theory into a broader theory on the structure of operators on a wide variety of mathematical spaces. It later found application in quantum mechanics.
Modern analysis includes Lebesgue Integration (far more widely applicable than Riemann) and the notion of Hilbert Spaces (henceforth abbreviated to H-Spaces).
H-Spaces are a generalisation of the notion of Euclidean space and used significantly in partial differential equations and quantum mechanics; the Sturm-Liouville theory, for example, which studies a particular second-order differential equation, utilises Hilbert spaces intrinsically.
H-Spaces have the very pleasant property that they are complete - any sequence of points in the space are guaranteed to converge to a point that actually lives within the space.
Spectral theory is another "advanced" analysis concept introduced by Mr Hilbert. It extends the eigenvalue-eigenvector theory into a broader theory on the structure of operators on a wide variety of mathematical spaces. It later found application in quantum mechanics.
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