Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
- The set where a linear inequality w·y ≤ β holds is closed in R^n, and so is the set where a whole family of such inequalities with a common bound holds.
- If every finite subfamily of a family of closed sets meets a compact set K in a common point, then the whole family meets K in a common point.
The Distance to a Subset of a Metric Space: Bounds, the Lipschitz Estimate, Continuity, and Vanishing Exactly on the Closure
lemmalem:distance-to-subset-metric-2026aAnalysisThe distance to a nonempty subset is a nonnegative lower bound of the point distances that they approximate, is 1-Lipschitz and hence continuous, vanishes exactly at the closure points, and is positive off a closed set.Distance from a Point to a Nonempty Subset of a Metric Space
definitiondef:distance-to-subset-metric-2026aAnalysisThe distance from a point to a nonempty subset of a metric space is the infimum of the distances from the point to the points of the subset.- A hyperplane supports a subset C of R^n at a point x of the closure of C if it passes through x and C lies in its closed half-space a·y ≤ α.
Separating Hyperplanes in : Separation, Strict Separation and Strong Separation
definitiondef:separating-hyperplane-rn-2026bAnalysisA hyperplane separates two subsets of R^n if they lie in opposite closed half-spaces, strictly if in opposite open half-spaces, and strongly if a positive margin separates them.- The hyperplane a·x = α in R^n for a nonzero normal a, with its two closed and two open half-spaces.
- Standing notation for separation and support of convex sets in R^n: the convex-sets setting, background facts on the dot product and the Euclidean norm, and the unit sphere.
The Continuous Image of a Compact Subset of a Metric Space is Compact
lemmalem:continuous-image-compact-metric-2026aA function that is continuous on a compact subset of a metric space maps it onto a compact subset of the target metric space.- On the line spanned by a nonzero vector x₀ of a real normed space, the functional t x₀ ↦ t c is linear and dominated by a sublinear functional p whenever -p(-x₀) ≤ c ≤ p(x₀).
- For compact subsets A and B of R^n, the set of all points (1-s)a+sb with a in A, b in B and s in [0,1] is compact.
Closure and Interior in a Metric Space: Containments, the Closure is Closed, the Interior is Open
lemmalem:closure-interior-metric-2026aFor a subset A of a metric space, int A is contained in A and A in cl A; cl A is closed and int A is open; A is closed exactly when cl A = A and open exactly when int A = A.- A point is a closure point of a subset of a metric space if every open ball around it meets the subset; the closure is the set of closure points.
- A point is an interior point of a subset of a metric space if some open ball around it lies in the subset; the interior is the set of interior points.
Strict Separation of a Closed Convex Set from a Compact Convex Set in a Separable Real Normed Space
theoremthm:strict-separation-separable-2026aAnalysisThe second geometric form of the Hahn–Banach theorem: in a separable real normed space, a closed convex set and a disjoint compact convex set, both nonempty, are strictly separated by a closed hyperplane; consequently a closed convex set is the set of points satisfying every line…- The convex hull of a compact subset of R^n is compact; in particular the convex hull of finitely many points is compact.
Separation of an Open Convex Set from a Disjoint Convex Set in a Separable Real Normed Space
theoremthm:separation-open-convex-separable-2026aAnalysisThe first geometric form of the Hahn–Banach theorem: in a separable real normed space, two disjoint nonempty convex sets, one of them open, are separated by a closed hyperplane F = α, with strict inequality on the open set.- Every point of the convex hull of a subset S of R^n is a convex combination of at most n+1 points of S.
Separating a Point from an Open Convex Set in a Separable Real Normed Space
lemmalem:separation-point-open-convex-2026aAnalysisIn a separable real normed space, a point outside a nonempty open convex set is strictly separated from it by a bounded linear functional.The Convex Hull Consists of the Convex Combinations of Finitely Many Points
theoremthm:convex-hull-combinations-rn-2026aAnalysisThe convex hull of a subset S of R^n is the set of all convex combinations of finitely many points of S.