Theorems

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

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Greatest Element of a Finite Family in a Totally Ordered Set
lemmalem:finite-family-greatest-element-2026bSet Theory
Let $A$ be a set equipped with a total order $\le$ , let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$ , and let $c\in A^{n}$ be an $n$ -tuple in $A$ , with components $c_{k}$ . Then there exists $j\in[n]$ such that $c_{k}\le c_{j}$ for every $k\in[n]$ .
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-03T00:13:44.079712Z
Self-Adjointness, Unitarity and Orthogonal Projections Through the Adjoint in Finite Dimensions
lemmalem:operator-classes-via-adjoint-2026dAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space that has an orthonormal basis $e\in V^{n}$ for some natural number $n$ , where $V^{n}$ is the set of $n$ -tuples in $V$ . By Uniqueness of the Adjoint, and Existence in Finite Dimensions every…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:57:11.365268Z
Every Linear Operator on a Space with a Finite Orthonormal Basis is Bounded
lemmalem:finite-orthonormal-basis-operator-bounded-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with induced norm $\lVert\cdot\rVert$ , which is a norm on $V$ by claim 2 of The Induced Norm is a Norm, and Induces a Metric. Let $n$ be a natural number, let $e\in V^{n}$ be an $n$ -tuple in $V$ th…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:55:50.113931Z
Properties of the Operator Norm
lemmalem:operator-norm-properties-2026aAnalysis Linear Algebra
Let $V$ be a complex vector space equipped with a norm $\lVert\cdot\rVert$ , let $S$ and $T$ be bounded linear operators on $V$ , and let $\lambda$ be a complex number with modulus $|\lambda|$ . Write $\lVert S\rVert_{\mathrm{op}}$ and $\lVert T\rVert_{\mathrm{op}}$ for their…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:54:33.169849Z
Operator Norm
definitiondef:operator-norm-2026aAnalysis Linear Algebra
Let $V$ be a complex vector space equipped with a norm $\lVert\cdot\rVert$ , let $T$ be a bounded linear operator on $V$ , and let $c$ be a real number. The number $c$ is an operator norm of $T$ if $c$ is a bound for $T$ and $c\le C\qquad\text{for every bound }C\text{ for }T,$ …
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:53:37.793066Z
Existence and Uniqueness of the Operator Norm
lemmalem:operator-norm-existence-uniqueness-2026aAnalysis Linear Algebra
Let $V$ be a complex vector space equipped with a norm $\lVert\cdot\rVert$ , and let $T$ be a bounded linear operator on $V$ . Let $B_{T}$ denote the set of those real numbers that are of the form $\lVert T(u)\rVert$ for some $u\in V$ with $\lVert u\rVert\le1$ , the order being that…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:53:34.030018Z
Bound for a Linear Operator and Bounded Linear Operator
definitiondef:bounded-linear-operator-2026aAnalysis Linear Algebra
Let $V$ be a complex vector space equipped with a norm $\lVert\cdot\rVert$ , let $T$ be a linear operator on $V$ , and let $C$ be a real number with $0\le C$ , the order being that of the ordered field of real numbers. The number $C$ is a bound for $T$ if…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:52:58.902093Z
Triangle Inequality for Finite Sums of Vectors
lemmalem:finite-sum-norm-triangle-2026aAnalysis Linear Algebra
Let $V$ be a complex vector space equipped with a norm $\lVert\cdot\rVert$ , let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$ , and let $v:[n]\to V$ be a map with values $v_{k}$ . Then…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:52:55.680935Z
Positive Definite Operator
definitiondef:positive-definite-operator-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 $V$ . The operator $T$ is positive definite if for every $u\in V$ the complex number $\langle u,T(u)\rangle$ is a real number satisfyin…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:52:50.972820Z
Properties of Unitary Operators
lemmalem:unitary-preserves-inner-product-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, with induced norm $\lVert\cdot\rVert$ , which is a norm on $V$ by claim 2 of The Induced Norm is a Norm, and Induces a Metric. Let $T$ be a unitary operator on $V$ . Throughout, $\lVert\cdot\rVert$ …
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:24:07.669932Z
Orthogonal Projection
definitiondef:orthogonal-projection-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, let $P$ be a linear operator on $V$ , and let the product of operators be as in that definition. The operator $P$ is an orthogonal projection if it is self-adjoint and satisfies $PP=P.$
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:23:28.754068Z
Positive Semi-Definite Operator
definitiondef:positive-semidefinite-operator-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $T$ be a linear operator on $V$ . The operator $T$ is positive semi-definite if for every $u\in V$ the complex number $\langle u,T(u)\rangle$ is a real number satisfying…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:23:24.689519Z
Unitary Operator
definitiondef:unitary-operator-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $T$ be a linear operator on $V$ . The operator $T$ is unitary if it is surjective, that is, every $v\in V$ satisfies $v=T(u)$ for some $u\in V$ , and if…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:23:21.164486Z
Self-Adjoint Operator
definitiondef:self-adjoint-operator-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $T$ be a linear operator on $V$ . The operator $T$ is self-adjoint if $\langle T(u),v\rangle=\langle u,T(v)\rangle\qquad\text{for all }u,v\in V.$
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:23:16.975158Z
Algebraic Properties of the Adjoint in Finite Dimensions
lemmalem:adjoint-properties-2026cAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space that has an orthonormal basis $e\in V^{n}$ for some natural number $n$ , where $V^{n}$ is the set of $n$ -tuples in $V$ . By Uniqueness of the Adjoint, and Existence in Finite Dimensions every…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:23:13.713069Z
Adjoint of a Linear Operator
definitiondef:adjoint-operator-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $S$ and $T$ be linear operators on $V$ . The operator $S$ is an adjoint of $T$ if $\langle S(u),v\rangle=\langle u,T(v)\rangle\qquad\text{for all }u,v\in V.$
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:22:27.867455Z
Uniqueness of the Adjoint, and Existence in Finite Dimensions
theoremthm:adjoint-existence-uniqueness-2026cAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with induced norm $\lVert\cdot\rVert$ , which is a norm on $V$ by claim 2 of The Induced Norm is a Norm, and Induces a Metric, and let $T$ be a linear operator on $V$ . Adjoints are as in…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:22:23.946910Z
Operations on Linear Operators
definitiondef:operator-operations-2026aAlgebra Linear Algebra
Let $K$ be a field and let $V$ be a vector space over $K$ . The set of linear operators on $V$ carries the following operations, each of which again yields a linear operator on $V$ by Sums, Scalar Multiples, Composites and the Identity are Linear Operators. Let $S$ and $T$ be line…
+0 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:21:39.836203Z
Sums, Scalar Multiples, Composites and the Identity are Linear Operators
lemmalem:operator-operations-linear-2026aAlgebra Linear Algebra
Let $K$ be a field, let $V$ be a vector space over $K$ , let $S$ and $T$ be linear operators on $V$ , and let $\lambda\in K$ . Then each of the following maps from $V$ to $V$ is a linear operator on $V$ . 1. (Sum) The map sending $u\in V$ to $S(u)+T(u)$ . 2. (Scalar multiple) The ma…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:21:36.186951Z
Linear Operator on a Vector Space
definitiondef:linear-operator-2026aAlgebra Linear Algebra
Let $K$ be a field and let $V$ be a vector space over $K$ . A linear operator on $V$ is a linear map from $V$ to $V$ .
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:21:31.809696Z

Showing 981-1000 of 1431

Back Next

Greatest Element of a Finite Family in a Totally Ordered Set

Self-Adjointness, Unitarity and Orthogonal Projections Through the Adjoint in Finite Dimensions

Every Linear Operator on a Space with a Finite Orthonormal Basis is Bounded

Properties of the Operator Norm

Operator Norm

Existence and Uniqueness of the Operator Norm

Bound for a Linear Operator and Bounded Linear Operator

Triangle Inequality for Finite Sums of Vectors

Positive Definite Operator

Properties of Unitary Operators

Orthogonal Projection

Positive Semi-Definite Operator

Unitary Operator

Self-Adjoint Operator

Algebraic Properties of the Adjoint in Finite Dimensions

Adjoint of a Linear Operator

Uniqueness of the Adjoint, and Existence in Finite Dimensions

Operations on Linear Operators

Sums, Scalar Multiples, Composites and the Identity are Linear Operators

Linear Operator on a Vector Space