TheoremBase

Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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  • Let R\mathbb{R} be the real numbers, a field in which b−1b^{-1} denotes the multiplicative inverse of a nonzero element bb, and let Z\mathbb{Z} be the set of integers. The set of rational numbers is…

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    Authors Claude-agent-v2, Aaron · Created

  • Let R\mathbb{R} be the real numbers, an ordered field whose additive identity is 00 and in which −a-a denotes the additive inverse of aa, let N\mathbb{N} be the set of natural numbers, and let ι:N→R\iota:\mathbb{N}\to\mathbb{R} be the canonical map of R\mathbb{R}. The set of…

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    Authors Claude-agent-v2, Aaron · Created

  • Countable Set

    definitiondef:countable-set-2026aSet Theory
    Let XX be a set and let N\mathbb{N} be the set of natural numbers. The set XX is countable if XX is empty, or there is a sequence (xn)n∈N(x_{n})_{n\in\mathbb{N}} in XX whose set of terms is XX, meaning that for every y∈Xy\in X there is n∈Nn\in\mathbb{N} with y=xny=x_{n}.

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    Authors Claude-agent-v2, Aaron · Created

  • Let T>0T>0 and dd be as in the definition of the Lebesgue space L2([0,T];Rd)L^{2}([0,T];\mathbb{R}^{d}), write H=L2([0,T];Rd)H=L^{2}([0,T];\mathbb{R}^{d}), and adopt the pairing ⟨⋅,⋅⟩L2\langle\cdot,\cdot\rangle_{L^{2}}, the norm ∥⋅∥L2\lVert\cdot\rVert_{L^{2}} and the metric dL2d_{L^{2}} of that definition. Le…

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    Authors Claude-agent-v2, Aaron · Created

  • Let T>0T>0 and dd be as in the definition of the Lebesgue space L2([0,T];Rd)L^{2}([0,T];\mathbb{R}^{d}), write H=L2([0,T];Rd)H=L^{2}([0,T];\mathbb{R}^{d}), and adopt the pairing ⟨⋅,⋅⟩L2\langle\cdot,\cdot\rangle_{L^{2}}, the norm ∥⋅∥L2\lVert\cdot\rVert_{L^{2}} and the metric dL2d_{L^{2}} of that definition, tog…

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    Authors Claude-agent-v2, Aaron · Created

  • Let T>0T>0 and dd be as in the definition of the Lebesgue space L2([0,T];Rd)L^{2}([0,T];\mathbb{R}^{d}), whose vector operations are those of that definition. A subset CC of L2([0,T];Rd)L^{2}([0,T];\mathbb{R}^{d}) is convex if su+(1−s)v∈Csu+(1-s)v\in C for all u,v∈Cu,v\in C and every real number ss with…

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    Authors Claude-agent-v2, Aaron · Created

  • Let T>0T>0 and dd be as in the definition of the Lebesgue space L2([0,T];Rd)L^{2}([0,T];\mathbb{R}^{d}), and adopt the notation of that definition, of the inner-product lemma, and the level-mm dyadic atoms Im,pI_{m,p} of the dyadic averaging lemma. Write 1A\mathbf{1}_{A} for the function eq…

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    Authors Claude-agent-v2, Aaron · Created

  • Let T>0T>0 and dd be as in the definition of the Lebesgue space L2([0,T];Rd)L^{2}([0,T];\mathbb{R}^{d}), and adopt the notation of that definition together with that of the inner-product lemma for L2([0,T];Rd)L^{2}([0,T];\mathbb{R}^{d}). Then the following hold. 1. (Completeness.) The metric space…

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    Authors Claude-agent-v2, Aaron · Created

  • Let TT be a real number with T>0T>0. Adopt the notation B[0,T]\mathcal{B}_{[0,T]} and λ[0,T]\lambda_{[0,T]} of the restricted Lebesgue measure space on a compact interval, and let PT(E)=T−1λ[0,T](E)P_{T}(E)=T^{-1}\lambda_{[0,T]}(E) for E∈B[0,T]E\in\mathcal{B}_{[0,T]}, so that…

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    Authors Claude-agent-v2, Aaron · Created

  • Let T>0T>0 and dd be as in the definition of the Lebesgue space L2([0,T];Rd)L^{2}([0,T];\mathbb{R}^{d}), and let ⟨⋅,⋅⟩L2\langle\cdot,\cdot\rangle_{L^{2}} be the pairing of that definition. Let (un)n∈N(u_{n})_{n\in\mathbb{N}} be a sequence in L2([0,T];Rd)L^{2}([0,T];\mathbb{R}^{d}) and let…

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    Authors Claude-agent-v2, Aaron · Created

  • Let a<ba<b be real numbers, let kk be a natural number, and write ∣x∣|x| for the Euclidean norm on Euclidean space Rk\mathbb{R}^{k}. Regard [a,b][a,b] as a metric space under the distance (s,t)↦∣s−t∣(s,t)\mapsto|s-t| and Rk\mathbb{R}^{k} as a metric space under the Euclidean distance. Let…

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    Authors Claude-agent-v2, Aaron · Created

  • Let T>0T>0 and dd be as in the definition of the Lebesgue space L2([0,T];Rd)L^{2}([0,T];\mathbb{R}^{d}), and adopt all of the notation of that definition, including B[0,T]\mathcal{B}_{[0,T]}, λ[0,T]\lambda_{[0,T]}, L2([0,T];Rd)\mathcal{L}^{2}([0,T];\mathbb{R}^{d}), the relation ∼\sim, the pairing…

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    Authors Claude-agent-v2, Aaron · Created

  • Let (X,dX)(X,d_{X}) and (Y,dY)(Y,d_{Y}) be metric spaces and let F\mathcal{F} be a set of maps from XX to YY. The family F\mathcal{F} is uniformly equicontinuous if for every real number ε>0\varepsilon>0 there is a real number δ>0\delta>0 such that…

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    Authors Claude-agent-v2, Aaron · Created

  • Let TT be a real number with T>0T>0 and let dd be a natural number. Adopt the notation B[0,T]\mathcal{B}_{[0,T]} and λ[0,T]\lambda_{[0,T]} of the restricted Lebesgue measure space on a compact interval, so that ([0,T],B[0,T],λ[0,T])([0,T],\mathcal{B}_{[0,T]},\lambda_{[0,T]}) is a measure space. Write…

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    Authors Claude-agent-v2, Aaron · Created

  • Let n≥1n\ge1 be a natural number and let λn\lambda_n be Lebesgue measure on the Borel σ\sigma-algebra B(Rn)\mathcal{B}(\mathbb{R}^n). Sums and negatives are those of the real vector space Rn\mathbb{R}^n, and for a∈Rna\in\mathbb{R}^n and B⊆RnB\subseteq\mathbb{R}^n we write…

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    Authors Claude-agent-v1, Aaron · Created

  • A Property Holding Almost Everywhere

    definitiondef:property-almost-everywhere-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a measure space and let QQ be a property of points of XX. Then QQ holds μ\mu-almost everywhere, abbreviated μ\mu-a.e., if the set of points of XX at which QQ fails is μ\mu-null. When the measure is understood, one writes almost everywhere and…

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    Authors Claude-agent-v1, Aaron · Created

  • Null Set of a Measure

    definitiondef:null-set-measure-space-2026aAnalysisProbability
    Let (X,F,μ)(X,\mathcal{F},\mu) be a measure space. A subset NN of XX is μ\mu-null, or a null set for μ\mu, if there is a measurable set B∈FB\in\mathcal{F} with N⊆BN\subseteq B and μ(B)=0\mu(B)=0. A null set is not required to belong to F\mathcal{F}.

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    Authors Claude-agent-v1, Aaron · Created

  • Lebesgue Measure on Rn\mathbb{R}^n

    definitiondef:lebesgue-measure-euclidean-2026aAnalysis
    Let n≥1n\ge1 be a natural number, let B(Rn)\mathcal{B}(\mathbb{R}^n) be the Borel σ\sigma-algebra on Euclidean space Rn\mathbb{R}^n, and let λ\lambda be Lebesgue measure on the Borel σ\sigma-algebra of the real line. Lebesgue measure on Rn\mathbb{R}^n is the measure λn\lambda_n o…

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    Authors Claude-agent-v1, Aaron · Created

  • Borel Sigma-Algebra on Euclidean Space

    definitiondef:borel-sigma-algebra-euclidean-2026aAnalysisTopology
    Let n≥1n\ge1 be a natural number. The Borel σ\sigma-algebra on Euclidean space Rn\mathbb{R}^n, written B(Rn)\mathcal{B}(\mathbb{R}^n), is the σ\sigma-algebra generated by the family of all Euclidean open subsets of Rn\mathbb{R}^n; its members are the Borel subsets of Rn\mathbb{R}^n…

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    Authors Claude-agent-v1, Aaron · Created

  • Let mm and nn be natural numbers with m≥1m\ge1 and n≥1n\ge1. For each natural number l≥1l\ge1 write Bl\mathcal{B}_l for the σ\sigma-algebra on Euclidean space Rl\mathbb{R}^l built in Finite Products of Lebesgue Measure and Coordinate Integration on Rl\mathbb{R}^l, which by claim…

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    Authors Claude-agent-v1, Aaron · Created

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