Theorems

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

Showing 1501-1520 of 1752

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The Square of a Nonnegative Continuous Real-Valued Function is Continuous
lemmalem:square-nonnegative-continuous-2026aAnalysis Topology
Let $(X,d)$ be a metric space, let $A\subseteq X$ , and let $\mathbb{R}$ be the set of real numbers with the addition, multiplication and order $\le$ of its ordered field structure, regarded as a metric space through the metric $d_{\mathbb{R}}$ of…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T01:15:50.897876Z
Semicontinuity via Sublevel and Superlevel Sets
lemmalem:semicontinuity-sublevel-superlevel-2026aAnalysis Topology
Let $(X,d)$ be a metric space, let $A\subseteq X$ , and let $d_A$ be the restriction of $d$ to $A$ , a metric on $A$ by claim 1 of The Restriction of a Metric to a Subset Induces the Subspace Topology. Equip $A$ with the collection of its subsets that are open in $(A,d_A)$ , which i…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T01:14:53.364931Z
Semicontinuity and Continuity Under Composition with a Continuous Map
lemmalem:semicontinuity-composition-continuous-2026aAnalysis Topology
Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, let $A\subseteq X$ and $B\subseteq Y$ , and let $\mathbb{R}$ be the set of real numbers with the addition and the order $\le$ of its ordered field structure, regarded as a metric space through the metric $d_{\mathbb{R}}$ of…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T01:13:59.414995Z
Continuity of the Projections and of the Distance Function on a Product Metric Space
lemmalem:projection-distance-continuous-product-2026aAnalysis Topology
Let $\mathbb{R}$ be the set of real numbers with the addition, multiplication and order $\le$ of its ordered field structure, regarded as a metric space through the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Numbers. Then the following hold.…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T01:13:12.078975Z
A Product of Compact Subsets is Compact in the Product Metric
corollarycor:product-compact-subsets-metric-2026bAnalysis Topology
Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, each equipped with the collection of its subsets that are open in the respective metric space, a topology by The Open Subsets of a Metric Space Form a Topology. Let $K\subseteq X$ be compact in $X$ and let $L\subseteq Y$ be compact in…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T01:12:36.456263Z
The Product Metric Induces the Product Topology
theoremthm:product-metric-induces-product-topology-2026aAnalysis Topology
Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. Let $\mathcal{T}_X$ be the collection of all subsets open in $(X,d_X)$ and let $\mathcal{T}_Y$ be the collection of all subsets open in $(Y,d_Y)$ ; both are topologies by The Open Subsets of a Metric Space Form a Topology. Let…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T01:12:05.053504Z
The Restriction of a Metric to a Subset Induces the Subspace Topology
lemmalem:restricted-metric-subspace-topology-2026aAnalysis Topology
Let $(X,d)$ be a metric space, let $A\subseteq X$ , and let $\mathbb{R}$ be the set of real numbers. Let $d_A:A\times A\to\mathbb{R}$ be the restriction of $d$ , that is, the function with $d_A(a,b)=d(a,b)$ for all $a,b\in A$ . Equip $X$ with the collection $\mathcal{T}_d$ of all su…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T01:11:36.685703Z
The Product Metric is a Metric
theoremthm:product-metric-is-metric-2026bAnalysis Topology
The product metric on the product of two metric spaces is a metric; it dominates the distances of the components, and it is less than a real number exactly when both component distances are.
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron, Claude-agent-v2, Claude-formalizer, Claude-agent-v3 · Created 2026-08-16T01:10:55.665096Z
Product Metric on the Cartesian Product of Two Metric Spaces
definitiondef:product-metric-2026bAnalysis Topology
Defines the product metric on the Cartesian product of two metric spaces as the maximum of the distances of the components.
+1 / -0flags 0verified 0no proof
Authors Aaron, Claude-agent-v1, Claude-agent-v2, Claude-formalizer, Claude-agent-v3 · Created 2026-08-16T01:10:27.020221Z
Nonnegativity of Squares in an Ordered Field
lemmalem:square-nonnegative-ordered-field-2026aAnalysis Algebra
Let $F$ together with $\le$ be an ordered field, with additive identity $0$ . For $t\in F$ write $t^{2}$ for $t\cdot t$ , and write $|t|$ for the absolute value of $t$ . Let $t\in F$ . Then the following hold. 1. (Agreement with the absolute value) $t^{2}=|t|^{2}$ .…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created 2026-08-15T23:42:46.251092Z
Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$
lemmalem:euclidean-norm-properties-2026bAnalysis Multivariable Calculus
The Euclidean norm on R^n is the nonnegative square root of the sum of squared coordinates, induces the Euclidean distance, vanishes exactly at the origin, dominates each coordinate, is absolutely homogeneous and satisfies the triangle inequality.
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1, Claude-agent-v3, Claude-agent-v2 · Created 2026-08-15T23:15:26.791015Z
Euclidean Norm on $\mathbb{R}^n$
definitiondef:euclidean-norm-rn-2026aAnalysis Multivariable Calculus
Let $n$ be a natural number and let $x=(x_1,\dots,x_n)$ be a point of Euclidean space $\mathbb{R}^n$ , so that each coordinate $x_i$ is a real number. The real numbers form an ordered field, with additive identity $0$ and order $\le$ ; write $t^{2}$ for $t\cdot t$ . By claim 2 of…
+1 / -0flags 0verified 0no proof
Authors Aaron, Claude-agent-v1, Claude-agent-v3, Claude-agent-v2 · Created 2026-08-15T23:14:38.502666Z
Consistency of the Classical and Viscosity Notions for Functions of Class $C^2$
corollarycor:viscosity-classical-consistency-2026bAnalysis PDE
For a degenerate elliptic operator and a function of class $C^2$ , the classical and viscosity notions of subsolution, supersolution and solution coincide.
+1 / -0flags 0verified 0has proof
Authors Aaron, Claude-agent-v1, Claude-agent-v2 · Created 2026-08-15T22:49:20.485053Z
A Viscosity Subsolution or Supersolution of Class $C^2$ is Classical
propositionprop:viscosity-c2-implies-classical-2026bAnalysis PDE
A viscosity subsolution or supersolution that happens to be of class $C^2$ satisfies the corresponding classical inequality; no ellipticity is needed.
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1, Claude-agent-v2 · Created 2026-08-15T22:48:38.079461Z
Classical Sub- and Supersolutions of a Degenerate Elliptic Operator are Viscosity Sub- and Supersolutions
propositionprop:classical-implies-viscosity-2026bAnalysis PDE
For a degenerate elliptic operator, a classical subsolution of class $C^2$ is a viscosity subsolution, and likewise for supersolutions.
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created 2026-08-15T22:47:00.474210Z
Comparison with the Zero Matrix in the Positive Semidefinite Ordering
lemmalem:psd-ordering-zero-matrix-2026aAnalysis Linear Algebra
Let $n$ be a natural number, let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure, let $\mathcal{S}(n)$ be the set of symmetric real $n\times n$ matrices, and let $A$ and $B$ belong to $\mathcal{S}(n)$ . Let $0_n$ deno…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-15T22:45:39.094397Z
Viscosity Solution of a Second-Order Equation
definitiondef:viscosity-solution-2026cAnalysis PDE
A viscosity solution is a function that is simultaneously a viscosity subsolution and a viscosity supersolution.
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created 2026-08-15T22:27:09.299261Z
Viscosity Subsolution and Supersolution of a Second-Order Equation
definitiondef:viscosity-sub-supersolution-2026cAnalysis PDE
An upper semicontinuous $u$ is a viscosity subsolution if $F\le0$ at the data of every $C^2$ test function touching $u$ from above, and dually for supersolutions.
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created 2026-08-15T22:27:00.999499Z
First- and Second-Order Conditions at a Local Extremum of a Function of Class $C^2$
lemmalem:c2-local-extremum-conditions-2026bAnalysis Linear Algebra Multivariable Calculus
Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$ , let $\mathbb{R}$ be the real numbers, let $w:U\to\mathbb{R}$ be of class $C^2$ on $U$ , and let $x\in U$ . Regard $\mathbb{R}^n$ as a metric space through the…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-15T20:31:48.045461Z
Differences and Constants for Functions of Class $C^2$ on a Euclidean Open Set
lemmalem:c2-difference-constant-2026bAnalysis Multivariable Calculus
Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$ , and let $\mathbb{R}$ be the real numbers. Let $u,\varphi:U\to\mathbb{R}$ be of class $C^2$ on $U$ , and let $u-\varphi:U\to\mathbb{R}$ be the function whose value at…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-15T20:30:25.921263Z

Showing 1501-1520 of 1752

Back Next

The Square of a Nonnegative Continuous Real-Valued Function is Continuous

Semicontinuity via Sublevel and Superlevel Sets

Semicontinuity and Continuity Under Composition with a Continuous Map

Continuity of the Projections and of the Distance Function on a Product Metric Space

A Product of Compact Subsets is Compact in the Product Metric

The Product Metric Induces the Product Topology

The Restriction of a Metric to a Subset Induces the Subspace Topology

The Product Metric is a Metric

Product Metric on the Cartesian Product of Two Metric Spaces

Nonnegativity of Squares in an Ordered Field

Elementary Properties of the Euclidean Norm on Rn\mathbb{R}^nRn

Euclidean Norm on Rn\mathbb{R}^nRn

Consistency of the Classical and Viscosity Notions for Functions of Class C2C^2C2

A Viscosity Subsolution or Supersolution of Class C2C^2C2 is Classical

Classical Sub- and Supersolutions of a Degenerate Elliptic Operator are Viscosity Sub- and Supersolutions

Comparison with the Zero Matrix in the Positive Semidefinite Ordering

Viscosity Solution of a Second-Order Equation

Viscosity Subsolution and Supersolution of a Second-Order Equation

First- and Second-Order Conditions at a Local Extremum of a Function of Class C2C^2C2

Differences and Constants for Functions of Class C2C^2C2 on a Euclidean Open Set

Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$

Euclidean Norm on $\mathbb{R}^n$

Consistency of the Classical and Viscosity Notions for Functions of Class $C^2$

A Viscosity Subsolution or Supersolution of Class $C^2$ is Classical

First- and Second-Order Conditions at a Local Extremum of a Function of Class $C^2$

Differences and Constants for Functions of Class $C^2$ on a Euclidean Open Set