SWhat changed
Can a table of distances come from points in a Hilbert space, where lengths and angles behave as in Euclidean geometry? In a separable setting, Schoenberg connects that question to the kernels exp(−s d²), for every positive s. These kernels must be positive semidefinite on every finite selection of points. The result links geometry to algebraic tests on distance tables and explains why suitable fractional powers of distances admit Hilbert-space embeddings.
Scientific scope
Can a table of distances come from points in a Hilbert space, where lengths and angles behave as in Euclidean geometry? In a separable setting, Schoenberg connects that question to the kernels exp(−s d²), for every positive s. These kernels must be positive semidefinite on every finite selection of points. The result links geometry to algebraic tests on distance tables and explains why suitable fractional powers of distances admit Hilbert-space embeddings. The original target Hilbert space is real separable sequence space. Distance symmetry, nonnegativity and zero diagonal are required; when distinct points have zero distance, identification is needed for an injective metric embedding. Positive definite means positive semidefinite finite forms, not strictly positive matrices. A single arbitrary scale is not the criterion: all positive scales, or a suitable sequence accumulating at zero, are required. For the stated real L^p, l^p and finite-dimensional spaces, 0<p≤2 and distance exponent 0<gamma≤p/2 are the admitted range; 0<p<1 before the power is not a metric. Isometric embedding is not a finite-dimensional efficient feature algorithm. The separate completely-monotone 1938 paper, spherical 1942 theory and general Fourier-measure representation are not included.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The original target Hilbert space is real separable sequence space. Distance symmetry, nonnegativity and zero diagonal are required; when distinct points have zero distance, identification is needed for an injective metric embedding. Positive definite means positive semidefinite finite forms, not strictly positive matrices. A single arbitrary scale is not the criterion: all positive scales, or a suitable sequence accumulating at zero, are required. For the stated real L^p, l^p and finite-dimensional spaces, 0<p≤2 and distance exponent 0<gamma≤p/2 are the admitted range; 0<p<1 before the power is not a metric. Isometric embedding is not a finite-dimensional efficient feature algorithm. The separate completely-monotone 1938 paper, spherical 1942 theory and general Fourier-measure representation are not included.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
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