Lengths are closed intervals. The idea of length {theory of content} {content theory} can extend to open intervals. For open intervals, sum of subintervals that enclose points has greatest lower bound {outer content} and sum of polygonal regions makes least upper bound {inner content}. If outer content is less than or equal to inner content, interval has content.
length
If inner content equals outer content, inner content is interval length for one dimension.
additive
For finite number of intervals, sum of disjoint sets with content is sum of set contents {additivity property}.
For adjoint spaces, map of vector space onto line can extend to map of space including line {Hahn-Banach theorem}.
In closed intervals (a,b) that have countably infinite interval sets, if a <= x <= b, and x is inside at least one interval, x is inside at least one interval of any finite interval set {Heine-Borel theorem} (Eduard Heine) [1821 to 1881].
Closed space subsets have two unique elements, one in subset and the other orthonormal to every element in subspace {projection theorem} {Riesz representation theorem}.
Main contribution to integral is from points at which derivative equals zero {stationary phase principle} {principle of stationary phase}.
In bounded infinite point sets, a point exists in which any neighborhood has set points {Weierstrass-Bolzano theorem}.
In intervals, analysis can find function-root sets {spectral theory}. Root index is less than its multiplicity. General spectral theory can be for symmetric kernels. Function kernels can have orthonormal eigenfunctions {Hilbert-Schmidt theorem}. Eigenfunction roots {eigenvalue, spectral theory} can be point, band, or continuous spectrum.
Spectral theory can generalize Volterra's method {spectral radius theorem}.
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Date Modified: 2022.0225