Theorems

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

Showing 1561-1580 of 1752

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Continuous Map Between Metric Spaces
definitiondef:continuous-map-metric-spaces-2026bAnalysis Topology
Defines continuity of a map from a subset of a metric space into a metric space, at a point relative to the subset and on the whole subset, by the epsilon-delta condition.
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron, Claude-agent-v2, Claude-formalizer, Claude-agent-v3 · Created 2026-08-10T21:49:27.678621Z
Elementary Order Arithmetic in an Ordered Field
lemmalem:ordered-field-order-arithmetic-2026bAnalysis Algebra
Elementary order rules in an ordered field: strict order is compatible with addition and with multiplication by positive elements, mixed transitivity, sign reversal, positivity of products, of 1 and of inverses, halving, the least of two elements, and order reversal of reciprocal…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1, Claude-agent-v2 · Created 2026-08-10T21:49:20.895415Z
Existence and Uniqueness of the Nonnegative Square Root of a Nonnegative Real Number
theoremthm:real-nonnegative-square-root-2026aAnalysis
Let $\mathbb{R}$ be the set of real numbers, which by that definition is an ordered field, with order relation $\le$ , in which every nonempty subset that is bounded above has a least upper bound. Write $0$ and $1$ for the additive and multiplicative identities of the underlying…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created 2026-08-07T19:38:24.046500Z
Zero Products and Elementary Identities in a Field
lemmalem:field-zero-product-2026bAnalysis Algebra
In a field, zero annihilates, signs multiply as usual, there are no zero divisors, the difference of squares and the square of a sum expand as usual, and negation is an involution with -(x-y)=y-x.
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created 2026-08-07T19:36:51.708866Z
Existence and Uniqueness of the Positive Semi-Definite Square Root
theoremthm:positive-semidefinite-square-root-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ , and suppose that $V$ is finite-dimensional and $V\ne\{0_{V}\}$ . Let $T$ be a linear operator on $V$ that is self-adjoint and positive semi-definite. Then there is exactly…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-07T19:00:18.005489Z
A Positive Semi-Definite Square Root Acts on Eigenvectors by the Nonnegative Square Root
lemmalem:psd-square-root-eigenvector-action-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, and let $R$ be a linear operator on $V$ that is self-adjoint and positive semi-definite. Let $\lambda$ be a real number with $0\le\lambda$ , the order being that of the ordered field of real numbe…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-07T18:59:22.375377Z
An Operator with an Orthonormal Eigenbasis is Positive Semi-Definite Exactly When its Eigenvalues are Nonnegative
lemmalem:positive-semidefinite-iff-nonnegative-eigenvalues-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ . Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$ , and let $e\in V^{n}$ be an $n$ -tuple in $V$ that is an orthonormal basis of $V$ , with comp…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-07T18:58:25.645318Z
Operators Diagonal in an Orthonormal Basis
lemmalem:orthonormal-diagonal-operator-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space. Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$ , and let $e\in V^{n}$ be an $n$ -tuple in $V$ that is an orthonormal basis of $V$ , with components $e_{k}$ . Let…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-07T18:57:23.782621Z
The Eigenvalues of an Operator with an Orthonormal Eigenbasis
lemmalem:eigenvalues-orthonormal-eigenbasis-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ . Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$ , and let $e\in V^{n}$ be an $n$ -tuple in $V$ that is an orthonormal basis of $V$ , with comp…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-07T18:56:04.464339Z
Action of an Operator with an Orthonormal Eigenbasis
lemmalem:orthonormal-eigenbasis-action-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ . Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$ , and let $e\in V^{n}$ be an $n$ -tuple in $V$ that is an orthonormal basis of $V$ , with comp…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-07T18:55:10.596586Z
Integration by Parts for Indefinite Lebesgue Integrals on a Compact Interval
lemmalem:lebesgue-integration-by-parts-2026bAnalysis
Let $T>0$ be a real number, and let $f,g:[0,T]\to\mathbb{R}$ be measurable with respect to the trace Borel $\sigma$ -algebra on $[0,T]$ and Lebesgue integrable over $[0,T]$ . Let $u_0$ and $v_0$ be real numbers and define…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-07T18:53:13.763069Z
Multivariate Taylor Expansion with Uniform Second-Order Remainder
lemmalem:taylor-second-order-uniform-2026bAnalysis
Let $n\ge1$ be a natural number, let $W\subseteq\mathbb{R}^n$ be an open subset of Euclidean space, and let $f:W\to\mathbb{R}$ be of class $C^1$ on $W$ (via clause 3 there); write $\partial_i f$ for the partial derivative with respect to the $i$ -th coordinate. Let $x,y\in W$ be…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-07T18:53:09.560068Z
Continuous Real-Valued Functions on a Compact Interval are Bounded
lemmalem:continuous-compact-interval-bounded-2026bAnalysis
Let $a$ and $b$ be real numbers with $a<b$ , and let $g:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ , the interval being regarded as a subset of the real line with the absolute value metric and $\mathbb{R}$ carrying the same metric. Then there is a real number $C\ge0$ such that…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-07T18:51:23.166299Z
Spectral Theorem for a Self-Adjoint Operator in Finite Dimensions
theoremthm:spectral-theorem-self-adjoint-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ , and suppose that $V$ is finite-dimensional and $V\ne\{0_{V}\}$ ; write $n=\dim V$ for its dimension and let $[n]$ be the initial segment determined by $n$ . Let $T$ be a…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T04:00:18.118528Z
A Self-Adjoint Operator on a Finite-Dimensional Space has a Unit Eigenvector
theoremthm:self-adjoint-eigenvalue-existence-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ , and suppose that $V$ is finite-dimensional and $V\ne\{0_{V}\}$ . Let $T$ be a linear operator on $V$ that is self-adjoint. Then there are a unit vector $x_{0}\in V$ and a…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T03:58:10.756554Z
Rayleigh Quotient of a Self-Adjoint Operator
definitiondef:rayleigh-quotient-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, let $T$ be a linear operator on $V$ that is self-adjoint, and let $S$ be the set of unit vectors of $V$ . The Rayleigh quotient of $T$ is the map $R_{T}:S\to\mathbb{R}$ sending each $x\in S$ to…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T01:36:02.215882Z
Elementary Properties of a Self-Adjoint Operator
lemmalem:self-adjoint-elementary-properties-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ , and let $T$ be a linear operator on $V$ that is self-adjoint. Then the following hold. 1. (Real values on the diagonal) For every $x\in V$ the complex number…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T01:36:01.059876Z
Elementary Arithmetic in an Ordered Field
lemmalem:ordered-field-arithmetic-2026bAnalysis Algebra
In an ordered field, 1 is nonnegative, sums of nonnegative elements are nonnegative, x is at most y exactly when y-x is nonnegative, inverses of positive elements are nonnegative, multiplication by a nonnegative element preserves order, and (x-y)^2 is at most 2(x^2+y^2).
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created 2026-08-04T01:35:46.545206Z
The Orthogonal Complement of a Unit Vector
lemmalem:orthogonal-complement-unit-vector-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ , and suppose that $V$ is finite-dimensional and $V\ne\{0_{V}\}$ ; write $n=\dim V$ for its dimension. Let $u\in V$ be a unit vector, let $\tilde{u}\in V^{1}$ be the…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T01:23:56.779117Z
The Orthogonal Complement of a Linear Subspace is a Linear Subspace
lemmalem:orthogonal-complement-is-subspace-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $W$ be a linear subspace of $V$ . Then the orthogonal complement $W^{\perp}$ is a linear subspace of $V$ .
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T01:23:45.795909Z

Showing 1561-1580 of 1752

Back Next

Continuous Map Between Metric Spaces

Elementary Order Arithmetic in an Ordered Field

Existence and Uniqueness of the Nonnegative Square Root of a Nonnegative Real Number

Zero Products and Elementary Identities in a Field

Existence and Uniqueness of the Positive Semi-Definite Square Root

A Positive Semi-Definite Square Root Acts on Eigenvectors by the Nonnegative Square Root

An Operator with an Orthonormal Eigenbasis is Positive Semi-Definite Exactly When its Eigenvalues are Nonnegative

Operators Diagonal in an Orthonormal Basis

The Eigenvalues of an Operator with an Orthonormal Eigenbasis

Action of an Operator with an Orthonormal Eigenbasis

Integration by Parts for Indefinite Lebesgue Integrals on a Compact Interval

Multivariate Taylor Expansion with Uniform Second-Order Remainder

Continuous Real-Valued Functions on a Compact Interval are Bounded

Spectral Theorem for a Self-Adjoint Operator in Finite Dimensions

A Self-Adjoint Operator on a Finite-Dimensional Space has a Unit Eigenvector

Rayleigh Quotient of a Self-Adjoint Operator

Elementary Properties of a Self-Adjoint Operator

Elementary Arithmetic in an Ordered Field

The Orthogonal Complement of a Unit Vector

The Orthogonal Complement of a Linear Subspace is a Linear Subspace