Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The -Envelopes on an Open Subset, under Penalisation of a Continuous Function, and on a Closed Subset
lemmalem:delta-envelopes-local-hilbert-triple-2026aAnalysisPDEThe -envelopes of a function on an open subset of a Hilbert triple are unchanged by restriction to a smaller open set; a continuous function penalised by is its own -envelope; and the -envelope minus a continuous function has closed superlevel…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-2026bAnalysisPDEThe delta-envelope u^-_delta is upper semicontinuous on V∩U, dual to u^+_delta under negation, has superlevel sets closed in (U, d_H), is dominated by C − delta h when u ≤ C, is monotone in u and antitone in delta with a gain of (delta' − delta) h, and equals u − delta h when u i…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 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…- Standing notation for a Hilbert triple: the two spaces and their subscripted notation, the standing separability hypothesis, the form operator with the Riesz and resolvent maps, the penalty function, open subsets of the larger space with their traces and the vocabulary used on th…
- Standing notation for real Hilbert spaces: numbers, maxima and minima, and sequences; the space; topology, including restricted metrics, balls, uniform continuity, moduli of continuity and restrictions of maps; bounded linear maps; bounded symmetric bilinear forms; projections; w…
Weak Compactness and Closedness of Bounded Subsets of the Small Space of a Hilbert Triple
lemmalem:hilbert-triple-closure-2026aAnalysisPDEIn a Hilbert triple (H,V,A), a V-bounded sequence has a subsequence converging weakly in V and in H to a point of V with the same norm bound; a V-bounded sequence converging in H has its limit in V with |x|_V <= R.The Resolvent of the Form Operator of a Hilbert Triple: Minimisation, Contraction, and Density of the Domain
theoremthm:hilbert-triple-resolvent-2026aAnalysisPDEFor alpha > 0 and x-bar in H the functional (1/2)|x|_V^2 + alpha |x - x-bar|_H^2 has a unique minimiser R_alpha x-bar on V, which lies in D(A) and solves A x + 2 alpha x = 2 alpha x-bar; R_alpha is an H-contraction, R_alpha x-bar -> x-bar in H as alpha -> infinity, and D(A) is de…Elementary Properties of a Hilbert Triple: the Embedding, the Riesz Map and the Form Operator
lemmalem:hilbert-triple-basic-2026aAnalysisPDEIn a Hilbert triple the inclusion V -> H is a bounded linear map carrying convergence and weak convergence from V to H; the Riesz map J: H -> V representing <z,.>_H on V is linear, injective and nonexpansive; D(A) is the range of J, A is linear and symmetric, and <Ax,x>_H = |x|_V…Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator
definitiondef:hilbert-triple-2026aAnalysisPDEA Hilbert triple (H,V,A) consists of separable real Hilbert spaces V inside H, V dense in H with |x|_H <= |x|_V, and the operator A defined on the set D(A) of x in V whose V-inner product against V is represented by an element Ax of H.