Theorems

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

Showing 801-820 of 1424

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Limits of Penalized Maxima on a Compact Set
theoremthm:penalization-limit-compact-2026bAnalysis Topology
Let $(X,d)$ be a metric space, equipped with the collection of its subsets that are open in $(X,d)$ , a topology by Metric Open Sets Form a Topology, and let $K\subseteq X$ be nonempty and compact in $X$ . Let $\mathbb{R}$ be the set of real numbers with the addition, multiplicatio…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T01:17:01.862465Z
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 Line. Then the following hold. 1. (Projections) L…
+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 Metric Open Sets Form a Topology. Let $K\subseteq X$ be compact in $X$ and let $L\subseteq Y$ be compact in $Y$ . Equip…
+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 Metric Open Sets Form a Topology. Let $d_{X\times Y}$ be the…
+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-2026aAnalysis Topology
Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces and let $d_{X\times Y}$ be the product metric on $X\times Y$ . Let $\mathbb{R}$ be the set of real numbers with the addition and the order $\le$ of its ordered field structure, and for $s,t\in\mathbb{R}$ write $s<t$ to mean that…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T01:10:55.665096Z
Product Metric on the Cartesian Product of Two Metric Spaces
definitiondef:product-metric-2026aAnalysis Topology
Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let $X\times Y$ be the Cartesian product of the sets $X$ and $Y$ . Let $\mathbb{R}$ be the set of real numbers with the order $\le$ of its ordered field structure, which is in particular a total order. The product metric on…
+1 / -0flags 0verified 0no proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T01:10:27.020221Z
Elementary Properties of the Maximum of Two Elements
lemmalem:maximum-two-elements-properties-2026aAlgebra Logic
Let $S$ be a set equipped with a total order $\le$ , let $a,b,c\in S$ , and let $\max\{a,b\}$ denote the maximum of $a$ and $b$ . Then the following hold. 1. (Upper bound) $a\le\max\{a,b\}$ and $b\le\max\{a,b\}$ . 2. (Attainment) $\max\{a,b\}=a$ or $\max\{a,b\}=b$ .…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-16T01:10:22.558946Z
Maximum of Two Elements of a Totally Ordered Set
definitiondef:maximum-two-elements-2026aAlgebra Logic
Let $S$ be a set equipped with a total order $\le$ , and let $a,b\in S$ . The maximum of $a$ and $b$ , written $\max\{a,b\}$ , is the element of $S$ defined as follows: if $a\le b$ , then $\max\{a,b\}$ is $b$ ; otherwise $\max\{a,b\}$ is $a$ .
+1 / -0flags 0verified 0no proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-16T01:10:18.422045Z
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 · Created 2026-08-15T23:42:46.251092Z
Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$
lemmalem:euclidean-norm-properties-2026aAnalysis Multivariable Calculus
Let $n$ be a natural number, let $x=(x_1,\dots,x_n)$ , $y$ and $h$ be points of Euclidean space $\mathbb{R}^n$ , and let $\lambda$ be a real number. The real numbers form an ordered field, with additive identity $0$ and order $\le$ ; for a real number $t$ write $t^2$ for $t\cdot t$ …
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-15T23:15:26.791015Z
Euclidean Space $\mathbb{R}^n$ is a Real Vector Space
propositionprop:rn-real-vector-space-2026aLinear Algebra Multivariable Calculus
Let $n$ be a natural number. Let $\mathbb{R}$ denote the real numbers, which form in particular a field, with additive identity $0$ , multiplicative identity $1$ , and additive inverse $-t$ of an element $t$ ; for $s,t\in\mathbb{R}$ write $s-t$ for $s+(-t)$ . Then Euclidean space…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-15T23:15:00.691113Z
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 · Created 2026-08-15T23:14:38.502666Z
The Origin of $\mathbb{R}^n$
definitiondef:origin-rn-2026aLinear Algebra Multivariable Calculus
Let $n$ be a natural number, and let $0$ denote the additive identity of the field of real numbers. The origin of Euclidean space $\mathbb{R}^n$ is the point $0_{\mathbb{R}^n}=(0,\dots,0)$ all of whose coordinates equal $0$ .
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-15T23:14:29.855597Z
Scalar Multiple of a Point of $\mathbb{R}^n$
definitiondef:scalar-multiple-rn-2026aLinear Algebra Multivariable Calculus
Let $n$ be a natural number, let $\lambda$ be a real number, and let $x=(x_1,\dots,x_n)$ be a point of Euclidean space $\mathbb{R}^n$ . The scalar multiple $\lambda x$ is the point of $\mathbb{R}^n$ defined by $\lambda x=(\lambda x_1,\dots,\lambda x_n),$ where in each coordin…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-15T23:14:27.401944Z
Sum of Points of $\mathbb{R}^n$
definitiondef:sum-rn-2026aLinear Algebra Multivariable Calculus
Let $n$ be a natural number, and let $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_n)$ be points of Euclidean space $\mathbb{R}^n$ , so that each $x_i$ and each $y_i$ is a real number. The sum $x+y$ is the point of $\mathbb{R}^n$ defined by $x+y=(x_1+y_1,\dots,x_n+y_n),$ where in e…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-15T23:14:23.845160Z
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

Showing 801-820 of 1424

Back Next

Limits of Penalized Maxima on a Compact Set

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

Elementary Properties of the Maximum of Two Elements

Maximum of Two Elements of a Totally Ordered Set

Nonnegativity of Squares in an Ordered Field

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

Euclidean Space Rn\mathbb{R}^nRn is a Real Vector Space

Euclidean Norm on Rn\mathbb{R}^nRn

The Origin of Rn\mathbb{R}^nRn

Scalar Multiple of a Point of Rn\mathbb{R}^nRn

Sum of Points of Rn\mathbb{R}^nRn

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

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

Euclidean Space $\mathbb{R}^n$ is a Real Vector Space

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

The Origin of $\mathbb{R}^n$

Scalar Multiple of a Point of $\mathbb{R}^n$

Sum of Points of $\mathbb{R}^n$

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