Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Quadratic Test Functions: a Global Majorant and Minorant Matching a Function to Second Order at a Point
lemmalem:taylor-test-function-hilbert-2026aAnalysisThe second-order Taylor polynomial of a function at a point, perturbed by , is of class on the whole space, majorises the function near the point when and minorises it when , and inherits its local maxima and minima.The Semicontinuous Envelopes of a Pointwise Maximum and of a Pointwise Minimum
lemmalem:envelope-maximum-metric-2026aAnalysisTopologyThe upper semicontinuous envelope of a pointwise maximum of two functions is the pointwise maximum of their upper envelopes, and dually for minima and lower envelopes.Local Bounds, Semicontinuous Envelopes and Local Extrema Are Unchanged on the Trace of an Open Set
lemmalem:envelopes-open-trace-metric-2026aAnalysisTopologyFor an open set meeting a set in a metric space, restriction to preserves local boundedness, gives the same semicontinuous envelopes at points of , and has the same local maxima and minima there.Classical Sub- and Supersolutions of a Degenerate Elliptic Equation on a Hilbert Triple are Viscosity Sub- and Supersolutions
propositionprop:classical-implies-viscosity-hilbert-triple-2026aAnalysisPDEFor a degenerate elliptic second-order equation operator on an open subset of a Hilbert triple, every classical subsolution (supersolution, solution) of class C^2 is a viscosity subsolution (supersolution, solution).Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of are Supersolutions of
lemmalem:viscosity-sign-reversal-hilbert-triple-2026aAnalysisPDEWith F̃(x,r,p,X) = −F(x,−r,−p,−X): F̃ is degenerate elliptic iff F is, the δ-shifts of F̃ are the negated δ-shifts of F at negated data, and u is a classical or viscosity subsolution of F iff −u is a classical or viscosity supersolution of F̃ (and conversely), so every statement…Basic Properties of the -Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity
lemmalem:delta-envelopes-basic-hilbert-triple-2026aAnalysisPDEThe δ-envelope u^-_δ is upper semicontinuous on V∩U, dual to u^+_δ under negation, has superlevel sets closed in (U, d_H), is dominated by C − δh when u ≤ C, is monotone in u and antitone in δ, and equals u − δh when u is upper semicontinuous; dual statements hold for u^+_δ.First- and Second-Order Conditions at a Local Extremum of a Function Penalised by on the Small Space
lemmalem:penalised-maximum-c2-hilbert-triple-2026aAnalysisPDEIf ψ is C^2 on an open subset U of H and ψ − λh has a local maximum on V∩U at x̂, then x̂ lies in D(A), λAx̂ = Dψ(x̂), and the restriction of D^2ψ(x̂) to V is at most λI_V; dually at a local minimum of ψ + λh.Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple
definitiondef:viscosity-sub-supersolution-hilbert-triple-2026aAnalysisPDEA locally bounded above function u on an open subset U of the large space is a viscosity subsolution of F if, for every δ > 0, every C^2 test function φ and every local maximum x̂ of u^-_δ − φ on V∩U, the shifted operator F^-_δ can be made ≤ ε at data in W×R×H×Sym(H) within ε of…Classical Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple
definitiondef:classical-sub-supersolution-hilbert-triple-2026aAnalysisPDEA function of class C^2 on an open subset U of the large space is a classical subsolution of F if F(x, u(x), Du(x), D^2u(x)|_V) ≤ 0 at every point of W = D(A)∩U, a classical supersolution if the reverse inequality holds, and a classical solution if it is both.The -Envelopes and of a Function on an Open Subset of a Hilbert Triple
definitiondef:delta-envelopes-hilbert-triple-2026aAnalysisPDEFor a function u on an open subset U of the large space of a Hilbert triple, locally bounded above (below), and δ > 0, the δ-envelopes are u^-_δ = (u − δh)^* and u^+δ = (u + δh)*, the semicontinuous envelopes taken on V∩U for the metric of H.Degenerate Elliptic Second-Order Equation Operator on a Hilbert Triple
definitiondef:degenerate-elliptic-hilbert-triple-2026aAnalysisPDEA second-order equation operator on a Hilbert triple is degenerate elliptic if it is nonincreasing in its form argument for the order on bounded symmetric bilinear forms on V.Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its -Shifts
definitiondef:second-order-operator-hilbert-triple-2026aAnalysisPDEA second-order equation operator on an open set U of the large space of a Hilbert triple is a real function of (x, r, p, X) with x in W = D(A)∩U, r real, p in H and X a bounded symmetric bilinear form on V. Its δ-shifts feed F the penalised arguments r ± δh(x), p ± δAx and Y|_V ±…The Penalty Function of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds
lemmalem:hilbert-triple-penalty-2026aAnalysisPDEFor the penalty function h(x) = (1/2)|x|_V^2 on the small space of a Hilbert triple: h dominates (1/2)|x|_H^2, expands exactly to second order with gradient x in V and Ax in H, has sublevel sets closed in H (so h is lower semicontinuous on V for the metric of H), D(A) meets every…A Linear Perturbation Producing a Sequentially Strict Maximum under a Coercive Bound
corollarycor:perturbed-maximum-linear-hilbert-2026aAnalysisPDEIf a function with closed superlevel sets on a subset of a real Hilbert space is dominated by a constant minus a positive multiple of the squared norm, then subtracting a linear functional of arbitrarily small norm makes it attain a sequentially strict maximum.A Perturbed Maximum Principle of Borwein-Preiss Type in a Real Hilbert Space
theoremthm:perturbed-maximum-hilbert-2026aAnalysisPDEA function bounded above with closed superlevel sets on a subset of a real Hilbert space becomes, after subtracting a single small quadratic, a function attaining a sequentially strict maximum at a point close to any given near-maximiser.The Subspaces Spanned by an Orthonormal Sequence and Exhausting Sequences
lemmalem:orthonormal-basis-exhausting-2026aAnalysisThe spans of the initial segments of an orthonormal sequence are finite-dimensional closed subspaces with an explicit orthogonal projection, and they form an exhausting sequence exactly when the orthonormal sequence is a basis.Orthonormal Expansions in a Real Hilbert Space
theoremthm:orthonormal-expansion-hilbert-2026aAnalysisBessel's inequality, the Riesz-Fischer criterion for convergence of an orthonormal series, and the equivalence of totality, the expansion of every vector, Parseval's identity and density of the finite linear combinations.- An orthonormal sequence in a real Hilbert space is an orthonormal basis if the only vector orthogonal to every member of the sequence is zero; completeness of the space is part of the definition.
The Doubling Form on the Product of a Real Hilbert Space with Itself
lemmalem:doubling-form-product-hilbert-2026aAnalysisPDEThe bilinear form sending a pair of points of the product to the inner product of their differences is a bounded symmetric bilinear form of norm at most two, and the corresponding multiple of the squared distance between coordinates is of class with explicit gradient and He…Properties of the Product of Two Real Inner Product Spaces
lemmalem:product-inner-product-space-2026aAnalysisThe norm of a pair, the comparison with the product metric, componentwise convergence and the Cauchy condition, boundedness of the coordinate maps and injections, and the inheritance of completeness and separability.