Complex abstract spaces {Hilbert space, analysis} can have infinite dimensions.
transformation
Coordinates can transform. Integral from x = a to x = b of K(x,y) * u(x) * dx can transform differentiable function u(x), where K(x,y) are differential equations.
purposes
Differential and integral equation eigenvalue theory is similar to n-dimensional-space linear transformations. Quantum-mechanics equations can use Hilbert-space spectral theory. Observed values are linear symmetric operators in Hilbert space. Linear symmetric energy-operator eigenvalues and eigenfunctions are energy levels. Eigenvalue differences show emitted-light frequencies. Lebesgue integrable functions {square summable function} make spaces similar to Hilbert spaces of sequences. Banach spaces {complete normed vector space} are generalized Hilbert spaces.
Mathematical Sciences>Calculus>Analysis>Space
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Date Modified: 2022.0224